### geometry

This is the 100th entry on this blog! It’s taken a while, but we’ve arrived at a meaningless but convenient milestone. This post constitutes Part III of the posts on the topics course which I shared with Susama Agarwala. In the first, I summarized the core idea in the series of lectures I did, which introduced toposes and sheaves, and explained how, at least for appropriate sites, sheaves can be thought of as generalized spaces. In the second, I described the guest lecture by John Huerta which described how supermanifolds can be seen as an example of that notion.

In this post, I’ll describe the machinery I set up as part of the context for Susama’s talks. The connections are a bit tangential, but it gives some helpful context for what’s to come. Namely, my last couple of lectures were on sheaves with structure, and derived categories. In algebraic geometry and elsewhere, derived categories are a common tool for studying spaces. They have a cohomological flavour, because they involve sheaves of complexes (or complexes of sheaves) of abelian groups. Having talked about the background of sheaves in Part I, let’s consider how these categories arise.

### Structured Sheaves and Internal Constructions in Toposes

The definition of a (pre)sheaf as a functor valued in $Sets$ is the basic one, but there are parallel notions for presheaves valued in categories other than $Sets$ – for instance, in Abelian groups, rings, simplicial sets, complexes etc. Abelian groups are particularly important for geometry/cohomology.

But for the most part, as long as the target category can be defined in terms of sets and structure maps (such as the multiplication map for groups, face maps for simplicial sets, or boundary maps in complexes), we can just think of these in terms of objects “internal to a category of sheaves”. That is, we have a definition of “abelian group object” in any reasonably nice category – in particular, any topos. Then the category of “abelian group objects in $Sh(\mathcal{T})$” is equivalent to a category of “abelian-group-valued sheaves on $\mathcal{T}$“, denoted $Sh((\mathcal{T},J),\mathbf{AbGrp})$. (As usual, I’ll omit the Grothendieck topology $J$ in the notation from now on, though it’s important that it is still there.)

Sheaves of abelian groups are supposed to generalize the prototypical example, namely sheaves of functions valued in abelian groups, (indeed, rings) such as $\mathbb{Z}$, $\mathbb{R}$, or $\mathbb{C}$.

To begin with, we look at the category $Sh(\mathcal{T},\mathbf{AbGrp})$, which amounts to the same as the category of abelian group objects in  $Sh(\mathcal{T})$. This inherits several properties from $\mathbf{AbGrp}$ itself. In particular, it’s an abelian category: this gives us that there is a direct sum for objects, a zero object, exact sequences split, all morphisms have kernels and cokernels, and so forth. These useful properties all hold because at each $U \in \mathcal{T}$, the direct sum of sheaves of abelian group just gives $(A \oplus A')(U) = A(U) \oplus A'(U)$, and all the properties hold locally at each $U$.

So, sheaves of abelian groups can be seen as abelian groups in a topos of sheaves $Sh(\mathcal{T})$. In the same way, other kinds of structures can be built up inside the topos of sheaves, and there are corresponding “external” point of view. One good example would be simplicial objects: one can talk about the simplicial objects in $Sh(\mathcal{T},\mathbf{Set})$, or sheaves of simplicial sets, $Sh(\mathcal{T},\mathbf{sSet})$. (Though it’s worth noting that since simplicial sets model infinity-groupoids, there are more sophisticated forms of the sheaf condition which can be applied here. But for now, this isn’t what we need.)

Recall that simplicial objects in a category $\mathcal{C}$ are functors $S \in Fun(\Delta^{op},\mathcal{C})$ – that is, $\mathcal{C}$-valued presheaves on $\Delta$, the simplex category. This $\Delta$ has nonnegative integers as its objects, and the morphisms from $n$ to $m$ are the order-preserving functions from $\{ 1, 2, \dots, n \}$ to $\{ 1, 2, \dots, m \}$. If $\mathcal{C} = \mathbf{Sets}$, we get “simplicial sets”, where $S(n)$ is the “set of $n$-dimensional simplices”. The various morphisms in $\Delta$ turn into (composites of) the face and degeneracy maps. Simplicial sets are useful because they are a good model for “spaces”.

Just as with abelian groups, simplicial objects in $Sh(\mathcal{T})$ can also be seen as sheaves on $\mathcal{T}$ valued in the category $\mathbf{sSet}$ of simplicial sets, i.e. objects of $Sh(\mathcal{T},\mathbf{sSet})$. These things are called, naturally, “simplicial sheaves”, and there is a rather extensive body of work on them. (See, for instance, the canonical book by Goerss and Jardine.)

This correspondence is just because there is a fairly obvious bunch of isomorphisms turning functors with two inputs into functors with one input returning another functor with one input:

$Fun(\Delta^{op} \times \mathcal{T}^{op},\mathbf{Sets}) \cong Fun(\Delta^{op}, Fun(\mathcal{T}^{op}, \mathbf{Sets}))$

and

$Fun(\Delta^{op} \times \mathcal{T}^{op},\mathbf{Sets}) \cong Fun(\mathcal{T}^{op},Fun(\Delta^{op},\mathbf{Sets})$

(These are all presheaf categories – if we put a trivial topology on $\Delta$, we can refine this to consider only those functors which are sheaves in every position, where we use a certain product topology on $\Delta \times \mathcal{T}$.)

Another relevant example would be complexes. This word is a bit overloaded, but here I’m referring to the sort of complexes appearing in cohomology, such as the de Rahm complex, where the terms of the complex are the sheaves of differential forms on a space, linked by the exterior derivative. A complex $X^{\bullet}$ is a sequence of Abelian groups with boundary maps $\partial^i : X^i \rightarrow X^{i+1}$ (or just $\partial$ for short), like so:

$\dots \rightarrow^{\partial} X^0 \rightarrow^{\partial} X^1 \rightarrow^{\partial} X^2 \rightarrow^{\partial} \dots$

with the property that $\partial^{i+1} \circ \partial^i = 0$. Morphisms between these are sequences of morphisms between the terms of the complexes $(\dots,f_0,f_1,f_2,\dots)$ where each $f_i : X^i \rightarrow Y^i$ which commute with all the boundary maps. These all assemble into a category of complexes $C^{\bullet}(\mathbf{AbGrp})$. We also have $C^{\bullet}_+$ and $C^{\bullet}_-$, the (full) subcategories of complexes where all the negative (respectively, positive) terms are trivial.

One can generalize this to replace $\mathbf{AbGrp}$ by any category enriched in abelian groups, which we need to make sense of the requirement that a morphism is zero. In particular, one can generalize it to sheaves of abelian groups. This is an example where the above discussion about internalization can be extended to more than one structure at a time: “sheaves-of-(complexes-of-abelian-groups)” is equivalent to “complexes-of-(sheaves-of-abelian-groups)”.

This brings us to the next point, which is that, within $Sh(\mathcal{T},\mathbf{AbGrp})$, the last two examples, simplicial objects and complexes, are secretly the same thing.

### Dold-Puppe Correspondence

The fact I just alluded to is a special case of the Dold-Puppe correspondence, which says:

Theorem: In any abelian category $\mathcal{A}$, the category of simplicial objects $Fun(\Delta^{op},\mathcal{A})$ is equivalent to the category of positive chain complexes $C^{\bullet}_+(\mathcal{A})$.

The better-known name “Dold-Kan Theorem” refers to the case where $\mathcal{A} = \mathbf{AbGrp}$. If $\mathcal{A}$ is a category of $\mathbf{AbGrp}$-valued sheaves, the Dold-Puppe correspondence amounts to using Dold-Kan at each $U$.

The point is that complexes have only coboundary maps, rather than a plethora of many different face and boundary maps, so we gain some convenience when we’re looking at, for instance, abelian groups in our category of spaces, by passing to this equivalent description.

The correspondence works by way of two maps (for more details, see the book by Goerss and Jardine linked above, or see the summary here). The easy direction is the Moore complex functor, $N : Fun(\Delta^{op},\mathcal{A} \rightarrow C^{\bullet}_+(\mathcal{A})$. On objects, it gives the intersection of all the kernels of the face maps:

$(NS)_k = \bigcap_{j=1}^{k-1} ker(d_i)$

The boundary map from this is then just $\partial_n = (-1)^n d_n$. This ends up satisfying the “boundary-squared is zero” condition because of the identities for the face maps.

The other direction is a little more complicated, so for current purposes, I’ll leave you to follow the references above, except to say that the functor $\Gamma$ from complexes to simplicial objects in $\mathcal{A}$ is defined so as to be adjoint to $N$. Indeed, $N$ and $\Gamma$ together form an adjoint equivalence of the categories.

### Chain Homotopies and Quasi-Isomorphisms

One source of complexes in mathematics is in cohomology theories. So, for example, there is de Rahm cohomology, where one starts with the complex with $\Omega^n(M)$ the space of smooth differential $n$-forms on some smooth manifold $M$, with the exterior derivatives as the coboundary maps. But no matter which complex you start with, there is a sequence of cohomology groups, because we have a sequence of cohomology functors:

$H^k : C^{\bullet}(\mathcal{A}) \rightarrow \mathcal{A}$

given by the quotients

$H^k(A^{\bullet}) = Ker(\partial_k) / Im(\partial_{k-1})$

That is, it’s the cocycles (things whose coboundary is zero), up to equivalence where cocycles are considered equivalent if their difference is a coboundary (i.e. something which is itself the coboundary of something else). In fact, these assemble into a functor $H^{\bullet} : C^{\bullet}(\mathcal{A}) \rightarrow C^{\bullet}(\mathcal{A})$, since there are natural transformations between these functors

$\delta^k(A^{\bullet}) : H^k(A^{\bullet} \rightarrow H^{k+1}(A^{\bullet})$

which just come from the restrictions of the $\partial^k$ to the kernel $Ker(\partial^k)$. (In fact, this makes the maps trivial – but the main point is that this restriction is well-defined on equivalence classes, and so we get an actual complex again.) The fact that we get a functor means that any chain map $f^{\bullet} : A^{\bullet} \rightarrow B^{\bullet}$ gives a corresponding $H^{\bullet}(f^{\bullet}) : H^{\bullet}(A^{\bullet}) \rightarrow H^{\bullet}(B^{\bullet})$.

Now, the original motivation of cohomology for a space, like the de Rahm cohomology of a manifold $M$, is to measure something about the topology of $M$. If $M$ is trivial (say, a contractible space), then its cohomology groups are all trivial. In the general setting, we say that $A^{\bullet}$ is acyclic if all the $H^k(A^{\bullet}) = 0$. But of course, this doesn’t mean that the chain itself is zero.

More generally, just because two complexes have isomorphic cohomology, doesn’t mean they are themselves isomorphic, but we say that $f^{\bullet}$ is a quasi-isomorphism if $H^{\bullet}(f^{\bullet})$ is an isomorphism. The idea is that, as far as we can tell from the information that coholomology detects, it might as well be an isomorphism.

Now, for spaces, as represented by simplicial sets, we have a similar notion: a map between spaces is a quasi-isomorphism if it induces an isomorphism on cohomology. Then the key thing is the Whitehead Theorem (viz), which in this language says:

Theorem: If $f : X \rightarrow Y$ is a quasi-isomorphism, it is a homotopy equivalence.

That is, it has a homotopy inverse $f' : Y \rightarrow X$, which means there is a homotopy $h : f' \circ f \rightarrow Id$.

What about for complexes? We said that in an abelian category, simplicial objects and complexes are equivalent constructions by the Dold-Puppe correspondence. However, the question of what is homotopy equivalent to what is a bit more complicated in the world of complexes. The convenience we gain when passing from simplicial objects to the simpler structure of complexes must be paid for it with a little extra complexity in describing what corresponds to homotopy equivalences.

The usual notion of a chain homotopy between two maps $f^{\bullet}, g^{\bullet} : A^{\bullet} \rightarrow B^{\bullet}$ is a collection of maps which shift degrees, $h^k : A^k \rightarrow B^{k-1}$, such that $f-g = \partial \circ h$. That is, the coboundary of $h$ is the difference between $f$ and $g$. (The “co” version of the usual intuition of a homotopy, whose ingoing and outgoing boundaries are the things which are supposed to be homotopic).

The Whitehead theorem doesn’t work for chain complexes: the usual “naive” notion of chain homotopy isn’t quite good enough to correspond to the notion of homotopy in spaces. (There is some discussion of this in the nLab article on the subject. That is the reason for…

### Derived Categories

Taking “derived categories” for some abelian category can be thought of as analogous, for complexes, to finding the homotopy category for simplicial objects. It compensates for the fact that taking a quotient by chain homotopy doesn’t give the same “homotopy classes” of maps of complexes as the corresponding operation over in spaces.

That is, simplicial sets, as a model category, know everything about the homotopy type of spaces: so taking simplicial objects in $\mathcal{C}$ is like internalizing the homotopy theory of spaces in a category $\mathcal{C}$. So, if what we’re interested in are the homotopical properties of spaces described as simplicial sets, we want to “mod out” by homotopy equivalences. However, we have two notions which are easy to describe in the world of complexes, which between them capture the notion “homotopy” in simplicial sets. There are chain homotopies and quasi-isomorphisms. So, naturally, we mod out by both notions.

So, suppose we have an abelian category $\mathcal{A}$. In the background, keep in mind the typical example where $\mathcal{A} = Sh( (\mathcal{T},J), \mathbf{AbGrp} )$, and even where $\mathcal{T} = TOP(X)$ for some reasonably nice space $X$, if it helps to picture things. Then the derived category of $\mathcal{A}$ is built up in a few steps:

1. Take the category $C^{\bullet} ( \mathcal{A} )$ of complexes. (This stands in for “spaces in $\mathcal{A}$” as above, although we’ve dropped the “$+$“, so the correct analogy is really with spectra. This is a bit too far afield to get into here, though, so for now let’s just ignore it.)
2. Take morphisms only up to homotopy equivalence. That is, define the equivalence relation with $f \sim g$ whenever there is a homotopy $h$ with $f-g = \partial \circ h$.  Then $K^{\bullet}(\mathcal{A}) = C^{\bullet}(\mathcal{A})/ \sim$ is the quotient by this relation.
3. Localize at quasi-isomorphisms. That is, formally throw in inverses for all quasi-isomorphisms $f$, to turn them into actual isomorphisms. The result is $D^{\bullet}(\mathcal{A})$.

(Since we have direct sums of complexes (componentwise), it’s also possible to think of the last step as defining $D^{\bullet}(\mathcal{A}) = K^{\bullet}(\mathcal{A})/N^{\bullet}(\mathcal{A})$, where $N^{\bullet}(\mathcal{A})$ is the category of acyclic complexes – the ones whose cohomology complexes are zero.)

Explicitly, the morphisms of $D^{\bullet}(\mathcal{A})$ can be thought of as “zig-zags” in $K^{\bullet}(\mathcal{A})$,

$X^{\bullet}_0 \leftarrow X^{\bullet}_1 \rightarrow X^{\bullet}_2 \leftarrow \dots \rightarrow X^{\bullet}_n$

where all the left-pointing arrows are quasi-isomorphisms. (The left-pointing arrows are standing in for their new inverses in $D^{\bullet}(\mathcal{A})$, pointing right.) This relates to the notion of a category of spans: in a reasonably nice category, we can always compose these zig-zags to get one of length two, with one leftward and one rightward arrow. In general, though, this might not happen.

Now, the point here is that this is a way of extracting “homotopical” or “cohomological” information about $\mathcal{A}$, and hence about $X$ if $\mathcal{A} = Sh(TOP(X),\mathbf{AbGrp})$ or something similar. In the next post, I’ll talk about Susama’s series of lectures, on the subject of motives. This uses some of the same technology described above, in the specific context of schemes (which introduces some extra considerations specific to that world). It’s aim is to produce a category (and a functor into it) which captures all the cohomological information about spaces – in some sense a universal cohomology theory from which any other can be found.

John Huerta visited here for about a week earlier this month, and gave a couple of talks. The one I want to write about here was a guest lecture in the topics course Susama Agarwala and I were teaching this past semester. The course was about topics in category theory of interest to geometry, and in the case of this lecture, “geometry” means supergeometry. It follows the approach I mentioned in the previous post about looking at sheaves as a kind of generalized space. The talk was an introduction to a program of seeing supermanifolds as a kind of sheaf on the site of “super-points”. This approach was first proposed by Albert Schwartz, though see, for instance, this review by Christophe Sachse for more about this approach, and this paper (comparing the situation for real and complex (super)manifolds) for more recent work.

It’s amazing how many geometrical techniques can be applied in quite general algebras once they’re formulated correctly. It’s perhaps less amazing for supermanifolds, in which commutativity fails in about the mildest possible way.  Essentially, the algebras in question split into bosonic and fermionic parts. Everything in the bosonic part commutes with everything, and the fermionic part commutes “up to a negative sign” within itself.

### Supermanifolds

Supermanifolds are geometric objects, which were introduced as a setting on which “supersymmetric” quantum field theories could be defined. Whether or not “real” physics has this symmetry (the evidence is still pending, though ), these are quite nicely behaved theories. (Throwing in extra symmetry assumptions tends to make things nicer, and supersymmetry is in some sense the maximum extra symmetry we might reasonably hope for in a QFT).

Roughly, the idea is that supermanifolds are spaces like manifolds, but with some non-commuting coordinates. Supermanifolds are therefore in some sense “noncommutative spaces”. Noncommutative algebraic or differential geometry start with various dualities to the effect that some category of spaces is equivalent to the opposite of a corresponding category of algebras – for instance, a manifold $M$ corresponds to the $C^{\infty}$ algebra $C^{\infty}(M,\mathbb{R})$. So a generalized category of “spaces” can be found by dropping the “commutative” requirement from that statement. The category $\mathbf{SMan}$ of supermanifolds only weakens the condition slightly: the algebras are $\mathbb{Z}_2$-graded, and are “supercommutative”, i.e. commute up to a sign which depends on the grading.

Now, the conventional definition of supermanifolds, as with schemes, is to say that they are spaces equipped with a “structure sheaf” which defines an appropriate class of functions. For ordinary (real) manifolds, this would be the sheaf assigning to an open set $U$ the ring $C^{\infty}(U,\mathbb{R})$ of all the smooth real-valued functions. The existence of an atlas of charts for the manifold amounts to saying that the structure sheaf locally looks like $C^{\infty}(V,\mathbb{R})$ for some open set $V \subset \mathbb{R}^p$. (For fixed dimension $p$).

For supermanifolds, the condition on the local rings says that, for fixed dimension $(p \bar q )$, a $p|q$-dimensional supermanifold has structure sheaf in which $they look like $\mathcal{O}(\mathcal{U}) \cong C^{\infty}(V,\mathbb{R}) \otimes \Lambda_q$ In this, $V$ is as above, and the notation $\Lambda_q = \Lambda ( \theta_1, \dots , \theta_q )$ refers to the exterior algebra, which we can think of as polynomials in the $\theta_i$, with the wedge product, which satisfies $\theta_i \wedge \theta_j = - \theta_j \wedge \theta_i$. The idea is that one is supposed to think of this as the algebra of smooth functions on a space with $p$ ordinary dimensions, and $q$ “anti-commuting” dimensions with coordinates $\theta_i$. The commuting variables, say $x_1,\dots,x_p$, are called “bosonic” or “even”, and the anticommuting ones are “fermionic” or “odd”. (The term “fermionic” is related to the fact that, in quantum mechanics, when building a Hilbert space for a bunch of identical fermions, one takes the antisymmetric part of the tensor product of their individual Hilbert spaces, so that, for instance, $v_1 \otimes v_2 = - v_2 \otimes v_1$). The structure sheaf picture can therefore be thought of as giving an atlas of charts, so that the neighborhoods locally look like “super-domains”, the super-geometry equivalent of open sets $V \subset \mathbb{R}^p$. In fact, there’s a long-known theorem of Batchelor which says that any real supermanifold is given exactly by the algebra of “global sections”, which looks like $\mathcal{O}(M) = C^{\infty}(M_{red},\mathbb{R}) \otimes \Lambda_q$. That is, sections in the local rings (“functions on” open neighborhoods of $M$) always glue together to give a section in $\mathcal{O}(M)$. Another way to put this is that every supermanifold can be seen as just bundle of exterior algebras. That is, a bundle over a base manifold $M_{red}$, whose fibres are the “super-points” $\mathbb{R}^{0|q}$ corresponding to $\Lambda_q$. The base space $M_{red}$ is called the “reduced” manifold. Any such bundle gives back a supermanifold, where the algebras in the structure sheaf are the algebras of sections of the bundle. One shouldn’t be too complacent about saying they are exactly the same, though: this correspondence isn’t functorial. That is, the maps between supermanifolds are not just bundle maps. (Also, Batchelor’s theorem works only for real, not for complex, supermanifolds, where only the local neighborhoods necessarily look like such bundles). Why, by the way, say that $\mathbb{R}^{0|q}$ is a super “point”, when $\mathbb{R}^{p|0}$ is a whole vector space? Since the fermionic variables are anticommuting, no term can have more than one of each $\theta_i$, so this is a finite-dimensional algebra. This is unlike $C{\infty}(V,\mathbb{R})$, which suggests that the noncommutative directions are quite different. Any element of $\Lambda_q$ is nilpotent, so if we think of a Taylor series for some function – a power series in the $(x_1,\dots,x_p,\theta_1,\dots,\theta_q)$ – we see note that no term has a coefficient for $\theta_i$ greater than 1, or of degree higher than $q$ in all the $\theta_i$ – so imagines that only infinitesimal behaviour in these directions exists at all. Thus, a supermanifold $M$ is like an ordinary $p$-dimensional manifold $M_{red}$, built from the ordinary domains $V$, equipped with a bundle whose fibres are a sort of “infinitesimal fuzz” about each point of the “even part” of the supermanifold, described by the $\Lambda_q$. But this intuition is a bit vague. We can sharpen it a bit using the functor of points approach… ### Supermanifolds as Manifold-Valued Sheaves As with schemes, there is also a point of view that sees supermanifolds as “ordinary” manifolds, constructed in the topos of sheaves over a certain site. The basic insight behind the picture of these spaces, as in the previous post, is based on the fact that the Yoneda lemma lets us think of sheaves as describing all the “probes” of a generalized space (actually an algebra in this case). The “probes” are the objects of a certain category, and are called “superpoints“. This category is just $\mathbf{Spt} = \mathbf{Gr}^{op}$, the opposite of the category of Grassman algebras (i.e. exterior algebras) – that is, polynomial algebras in noncommuting variables, like $\Lambda(\theta_1,\dots,\theta_q)$. These objects naturally come with a $\mathbb{Z}_2$-grading, which are spanned, respectively, by the monomials with even and odd degree: $\Lambda_q =$latex \mathbf{SMan}$ (\Lambda_q)_0 \oplus (\Lambda_q)_1$$(\Lambda_q)_0 = span( 1, \theta_i \theta_j, \theta_{i_1}\dots\theta{i_4}, \dots )$ and $(\Lambda_q)_1 = span( \theta_i, \theta_i \theta_j \theta_k, \theta_{i_1}\dots\theta_{i_5},\dots )$ This is a $\mathbb{Z}_2$-grading since the even ones commute with anything, and the odd ones anti-commute with each other. So if $f_i$ and $f_j$ are homogeneous (live entirely in one grade or the other), then $f_i f_j = (-1)^{deg(i)deg(j)} f_j f_i$. The $\Lambda_q$ should be thought of as the $(0|q)$-dimensional supermanifold: it looks like a point, with a $q$-dimensional fermionic tangent space (the “infinitesimal fuzz” noted above) attached. The morphisms in $\mathbf{Spt}$ from $\Lambda_q$ to$llatex \Lambda_r$are just the grade-preserving algebra homomorphisms from $\Lambda_r$ to $\Lambda_q$. There are quite a few of these: these objects are not terminal objects like the actual point. But this makes them good probes. Thi gets to be a site with the trivial topology, so that all presheaves are sheaves. Then, as usual, a presheaf $M$ on this category is to be understood as giving, for each object $A=\Lambda_q$, the collection of maps from $\Lambda_q$ to a space $M$. The case $q=0$ gives the set of points of $M$, and the various other algebras $A$ give sets of “$A$-points”. This term is based on the analogy that a point of a topological space (or indeed element of a set) is just the same as a map from the terminal object $1$, the one point space (or one element set). Then an “$A$-point” of a space $X$ is just a map from another object $A$. If $A$ is not terminal, this is close to the notion of a “subspace” (though a subspace, strictly, would be a monomorphism from $A$). These are maps from $A$ in $\mathbf{Spt} = \mathbf{Gr}^{op}$, or as algebra maps, $M_A$ consists of all the maps $\mathcal{O}(M) \rightarrow A$. What’s more, since this is a functor, we have to have a system of maps between the $M_A$. For any algebra maps $A \rightarrow A'$, we should get corresponding maps $M_{A'} \rightarrow M_A$. These are really algebra maps $\Lambda_q \rightarrow \Lambda_{q'}$, of which there are plenty, all determined by the images of the generators $\theta_1, \dots, \theta_q$. Now, really, a sheaf on $\mathbf{Spt}$ is actually just what we might call a “super-set”, with sets $M_A$ for each $A \in \mathbf{Spt}$. To make super-manifolds, one wants to say they are “manifold-valued sheaves”. Since manifolds themselves don’t form a topos, one needs to be a bit careful about defining the extra structure which makes a set a manifold. Thus, a supermanifold $M$ is a manifold constructed in the topos $Sh(\mathbf{Spt})$. That is, $M$ must also be equipped with a topology and a collection of charts defining the manifold structure. These are all construed internally using objects and morphisms in the category of sheaves, where charts are based on super-domains, namely those algebras which look like $C^{\infty}(V) \otimes \Lambda_q$, for $V$ an open subset of $\mathbb{R}^p$. The reduced manifold $M_{red}$ which appears in Batchelor’s theorem is the manifold of ordinary points $M_{\mathbb{R}}$. That is, it is all the $\mathbb{R}$-points, where $\mathbb{R}$ is playing the role of functions on the zero-dimensional domain with just one point. All the extra structure in an atlas of charts for all of $M$ to make it a supermanifold amounts to putting the structure of ordinary manifolds on the $M_A$ – but in compatible ways. (Alternatively, we could have described $\mathbf{SMan}$ as sheaves in $Sh(\mathbf{SDom})$, where $\mathbf{SDom}$ is a site of “superdomains”, and put all the structure defining a manifold into $\mathbf{SDom}$. But working over super-points is preferable for the moment, since it makes it clear that manifolds and supermanifolds are just manifestations of the same basic definition, but realized in two different toposes.) The fact that the manifold structure on the $M_A$ must be put on them compatibly means there is a relatively nice way to picture all these spaces. ### Values of the Functor of Points as Bundles The main idea which I find helps to understand the functor of points is that, for every superpoint $\mathbb{R}^{0|n}$ (i.e. for every Grassman algebra $A=\Lambda_n$), one gets a manifold $M_A$. (Note the convention that $q$ is the odd dimension of $M$, and $n$ is the odd dimension of the probe superpoint). Just as every supermanifold is a bundle of superpoints, every manifold $M_A$ is a perfectly conventional vector bundle over the conventional manifold $M_{red}$ of ordinary points. So for each $A$, we get a bundle, $M_A \rightarrow M_{red}$. Now this manifold, $M_{red}$, consists exactly of all the “points” of $M$ – this tells us immediately that $\mathbf{SMan}$ is not a category of concrete sheaves (in the sense I explained in the previous post). Put another way, it’s not a concrete category – that would mean that there is an underlying set functor, which gives a set for each object, and that morphisms are determined by what they do to underlying sets. Non-concrete categories are, by nature, trickier to understand. However, the functor of points gives a way to turn the non-concrete $M$ into a tower of concrete manifolds $M_A$, and the morphisms between various $M$ amount to compatible towers of maps between the various $M_A$ for each $A$. The fact that the compatibility is controlled by algebra maps $\Lambda_q \rightarrow \Lambda_{q'}$ explains why this is the same as maps between these bundles of superpoints. Specifically, then, we have $M_A = \{ \mathcal{O}(M) \rightarrow A \}$ This splits into maps of the even parts, and of the odd parts, where the grassman algebra $A = \Lambda_n$ has even and odd parts: $A = A_0 \oplus A_1$, as above. Similarly, $\mathcal{O}(M)$ splits into odd and even parts, and since the functions on $M_{red}$ are entirely even, this is: $( \mathcal{O}(M))_0 = C^{\infty}(M_{red}) \otimes ( \Lambda_q)_0$ and $( \mathcal{O}(M))_1 = C^{\infty}(M_{red}) \otimes (\Lambda_q)_1)$ Now, the duality of “hom” and tensor means that $Hom(\mathcal{O}(M),A) \cong \mathcal{O}(M) \otimes A$, and algebra maps preserve the grading. So we just have tensor products of these with the even and odd parts, respectively, of the probe superpoint. Since the even part $A_0$ includes the multiples of the constants, part of this just gives a copy of $U$ itself. The remaining part of $A_0$ is nilpotent (since it’s made of even-degree polynomials in the nilpotent $\theta_i$, so what we end up with, looking at the bundle over an open neighborhood $U \subset M_{red}$, is: $U_A = U \times ( (\Lambda_q)_0 \otimes A^{nil}_0) \times ((\Lambda_q)_1 \otimes A_1)$ The projection map $U_A \rightarrow U$ is the obvious projection onto the first factor. These assemble into a bundle over $M_{red}$. We should think of these bundles as “shifting up” the nilpotent part of $M$ (which are invisible at the level of ordinary points in $M_{red}$) by the algebra $A$. Writing them this way makes it clear that this is functorial in the superpoints $A = \Lambda_n$: given choices $n$ and $n'$, and any morphism between the corresponding $A$ and $A'$, it’s easy to see how we get maps between these bundles. Now, maps between supermanifolds are the same thing as natural transformations between the functors of points. These include maps of the base manifolds, along with maps between the total spaces of all these bundles. More, this tower of maps must commute with all those bundle maps coming from algebra maps $A \rightarrow A'$. (In particular, since $A = \Lambda_0$, the ordinary point, is one of these, they have to commute with the projection to $M_{red}$.) These conditions may be quite restrictive, but it leaves us with, at least, a quite concrete image of what maps of supermanifolds ### Super-Poincaré Group One of the main settings where super-geometry appears is in so-called “supersymmetric” field theories, which is a concept that makes sense when fields live on supermanifolds. Supersymmetry, and symmetries associated to super-Lie groups, is exactly the kind of thing that John has worked on. A super-Lie group, of course, is a supermanifold that has the structure of a group (i.e. it’s a Lie group in the topos of presheaves over the site of super-points – so the discussion above means it can be thought of as a big tower of Lie groups, all bundles over a Lie group $G_{red}$). In fact, John has mostly worked with super-Lie algebras (and the connection between these and division algebras, though that’s another story). These are $\mathbb{Z}_2$-graded algebras with a Lie bracket whose commutation properties are the graded version of those for an ordinary Lie algebra. But part of the value of the framework above is that we can simply borrow results from Lie theory for manifolds, import it into the new topos $PSh(\mathbf{Spt})$, and know at once that super-Lie algebras integrate up to super-Lie groups in just the same way that happens in the old topos (of sets). Supersymmetry refers to a particular example, namely the “super-Poincaré group”. Just as the Poincaré group is the symmetry group of Minkowski space, a 4-manifold with a certain metric on it, the super-Poincaré group has the same relation to a certain supermanifold. (There are actually a few different versions, depending on the odd dimension.) The algebra is generated by infinitesimal translations and boosts, plus some “translations” in fermionic directions, which generate the odd part of the algebra. Now, symmetry in a quantum theory means that this algebra (or, on integration, the corresponding group) acts on the Hilbert space $\mathcal{H}$ of possible states of the theory: that is, the space of states is actually a representation of this algebra. In fact, to make sense of this, we need a super-Hilbert space (i.e. a graded one). The even generators of the algebra then produce grade-preserving self-maps of $\mathcal{H}$, and the odd generators produce grade-reversing ones. (This fact that there are symmetries which flip the “bosonic” and “fermionic” parts of the total $\mathcal{H}$ is why supersymmetric theories have “superpartners” for each particle, with the opposite parity, since particles are labelled by irreducible representations of the Poincaré group and the gauge group). To date, so far as I know, there’s no conclusive empirical evidence that real quantum field theories actually exhibit supersymmetry, such as detecting actual super-partners for known particles. Even if not, however, it still has some use as a way of developing toy models of quite complicated theories which are more tractable than one might expect, precisely because they have lots of symmetry. It’s somewhat like how it’s much easier to study computationally difficult theories like gravity by assuming, for instance, spherical symmetry as an extra assumption. In any case, from a mathematician’s point of view, this sort of symmetry is just a particularly simple case of symmetries for theories which live on noncommutative backgrounds, which is quite an interesting topic in its own right. As usual, physics generates lots of math which remains both true and interesting whether or not it applies in the way it was originally suggested. In any case, what the functor-of-points viewpoint suggests is that ordinary and super- symmetries are just two special cases of “symmetries of a field theory” in two different toposes. Understanding these and other examples from this point of view seems to give a different understanding of what “symmetry”, one of the most fundamental yet slippery concepts in mathematics and science, actually means. This semester, Susama Agarwala and I have been sharing a lecture series for graduate students. (A caveat: there are lecture notes there, by student request, but they’re rough notes, and contain some mistakes, omissions, and represent a very selective view of the subject.) Being a “topics” course, it consists of a few different sections, loosely related, which revolve around the theme of categorical tools which are useful for geometry (and topology). What this has amounted to is: I gave a half-semester worth of courses on toposes, sheaves, and the basics of derived categories. Susama is now giving the second half, which is about motives. This post will talk about the part of the course I gave. Though this was a whole series of lectures which introduced all these topics more or less carefully, I want to focus here on the part of the lecture which built up to a discussion of sheaves as spaces. Nothing here, or in the two posts to follow, is particularly new, but they do amount to a nice set of snapshots of some related ideas. Coming up soon: John Huerta is currently visiting Hamburg, and on July 8, he gave a guest-lecture which uses some of this machinery to talk about supermanifolds, which will be the subject of the next post in this series. In a later post, I’ll talk about Susama’s lectures about motives and how this relates to the discussion here (loosely). ### Grothendieck Toposes The first half of our course was about various aspects of Grothendieck toposes. In the first lecture, I talked about “Elementary” (or Lawvere-Tierney) toposes. One way to look at these is to say that they are categories $\mathcal{E}$ which have all the properties of the category of Sets which make it useful for doing most of ordinary mathematics. Thus, a topos in this sense is a category with a bunch of properties – there are various equivalent definitions, but for example, toposes have all finite limits (in particular, products), and all colimits. More particularly, they have “power objects”. That is, if $A$ and $B$ are objects of $\mathcal{E}$, then there is an object $B^A$, with an “evaluation map” $B^A \times A \rightarrow B$, which makes it possible to think of $B^A$ as the object of “morphisms from A to B”. The other main thing a topos has is a “subobject classifier”. Now, a subobject of $A \in \mathcal{E}$ is an equivalence class of monomorphisms into $A$ – think of sets, where this amounts to specifying the image, and the monomorphisms are the various inclusions which pick out the same subset as their image. A classifier for subobjects should be thought of as something like the two-element set is $Sets$, whose elements we can tall “true” and “false”. Then every subset of $A$ corresponds to a characteristic function $A \rightarrow \mathbf{2}$. In general, a subobject classifies is an object $\Omega$ together with a map from the terminal object, $T : 1 \rightarrow \Omega$, such that every inclusion of subobject is a pullback of $T$ along a characteristic function. Now, elementary toposes were invented chronologically later than Grothendieck toposes, which are a special class of example. These are categories of sheaves on (Grothendieck) sites. A site is a category $\mathcal{T}$ together with a “topology” $J$, which is a rule which, for each $U \in \mathcal{T}$, picks out $J(U)$, a set of collections of maps into $U$, called seives for $U$. They collections $J(U)$ have to satisfy certain conditions, but the idea can be understood in terms of the basic example, $\mathcal{T} = TOP(X)$. Given a topological space, $TOP(X)$ is the category whose objects are the open sets $U \subset X$, and the morphisms are all the inclusions. Then that each collection in $J(U)$ is an open cover of $U$ – that is, a bunch of inclusions of open sets, which together cover all of $U$ in the usual sense. (This is a little special to $TOP(X)$, where every map is an inclusion – in a general site, the $J(U)$ need to be closed under composition with any other morphism (like an ideal in a ring). So for instance, $\mathcal{T} = Top$, the category of topological spaces, the usual choice of $J(U)$ consists of all collections of maps which are jointly surjective.) The point is that a presheaf on $\mathcal{T}$ is just a functor $\mathcal{T}^{op} \rightarrow Sets$. That is, it’s a way of assigning a set to each $U \in \mathcal{T}$. So, for instance, for either of the cases we just mentioned, one has $B : \mathcal{T}^{op} \rightarrow Sets$, which assigns to each open set $U$ the set of all bounded functions on $U$, and to every inclusion the restriction map. Or, again, one has $C : \mathcal{T}^{op} \rightarrow Sets$, which assigns the set of all continuous functions. These two examples illustrate the condition which distinguishes those presheaves $S$ which are sheaves – namely, those which satisfy some “gluing” conditions. Thus, suppose we’re, given an open cover $\{ f_i : U_i \rightarrow U \}$, and a choice of one element $x_i$ from each $S(U_i)$, which form a “matching family” in the sense that they agree when restricted to any overlaps. Then the sheaf condition says that there’s a unique “amalgamation” of this family – that is, one element $x \in S(U)$ which restricts to all the $x_i$ under the maps $S(f_i) : S(U) \rightarrow S(U_i)$. ### Sheaves as Generalized Spaces There are various ways of looking at sheaves, but for the purposes of the course on categorical methods in geometry, I decided to emphasize the point of view that they are a sort of generalized spaces. The intuition here is that all the objects and morphisms in a site $\mathcal{T}$ have corresponding objects and morphisms in $Psh(\mathcal{T})$. Namely, the objects appear as the representable presheaves, $U \mapsto Hom(-,U)$, and the morphisms $U \rightarrow V$ show up as the induced natural transformations between these functors. This map $y : \mathcal{T} \rightarrow Psh(\mathcal{T})$ is called the Yoneda embedding. If $\mathcal{T}$ is at all well-behaved (as it is in all the examples we’re interested in here), these presheaves will always be sheaves: the image of $y$ lands in $Sh(\mathcal{T})$. In this case, the Yoneda embedding embeds $\mathcal{T}$ as a sub-category of $Sh(\mathcal{T})$. What’s more, it’s a full subcategory: all the natural transformations between representable presheaves come from the morphisms of $\mathcal{T}$-objects in a unique way. So $Sh(\mathcal{T})$ is, in this sense, a generalization of $\mathcal{T}$ itself. More precisely, it’s the Yoneda lemma which makes sense of all this. The idea is to start with the way ordinary $\mathcal{T}$-objects (from now on, just call them “spaces”) $S$ become presheaves: they become functors which assign to each $U$ the set of all maps into $S$. So the idea is to turn this around, and declare that even non-representable sheaves should have the same interpretation. The Yoneda Lemma makes this a sensible interpretation: it says that, for any presheaf $F \in Psh(\mathcal{T})$, and any $U \in \mathcal{T}$, the set $F(U)$ is naturally isomorphic to $Hom(y(U),F)$: that is, $F(U)$ literally is the collection of morphisms from $U$ (or rather, its image under the Yoneda embedding) and a “generalized space” $F$. (See also Tom Leinster’s nice discussion of the Yoneda Lemma if this isn’t familiar.) We describe $U$ as a “probe” object: one probes the space $F$ by mapping $U$ into it in various ways. Knowing the results for all $U \in \mathcal{T}$ tells you all about the “space” $F$. (Thus, for instance, one can get all the information about the homotopy type of a space if you know all the maps into it from spheres of all dimensions up to homotopy. So spheres are acting as “probes” to reveal things about the space.) Furthermore, since $Sh(\mathcal{T})$ is a topos, it is often a nicer category than the one you start with. It has limits and colimits, for instance, which the original category might not have. For example, if the kind of spaces you want to generalize are manifolds, one doesn’t have colimits, such as the space you get by gluing together two lines at a point. The sheaf category does. Likewise, the sheaf category has exponentials, and manifolds don’t (at least not without the more involved definitions needed to allow infinite-dimensional manifolds). These last remarks about manifolds suggest the motivation for the first example… ### Diffeological Spaces The lecture I gave about sheaves as spaces used this paper by John Baez and Alex Hoffnung about “smooth spaces” (they treat Souriau’s diffeological spaces, and the different but related Chen spaces in the same framework) to illustrate the point. They describe In that case, the objects of the sites are open (or, for Chen spaces, convex) subsets of $\mathbb{R}^n$, for all choices of $n$, the maps are the smooth maps in the usual sense (i.e. the sense to be generalized), and the covers are jointly surjective collections of maps. Now, that example is a somewhat special situation: they talk about concrete sheaves, on concrete sites, and the resulting categories are only quasitoposes – a slightly weaker condition than being a topos, but one still gets a useful collection of spaces, which among other things include all manifolds. The “concreteness” condition – that $\mathcal{T}$ has a terminal object to play the role of “the point”. Being a concrete sheaf then means that all the “generalized spaces” have an underlying set of points (namely, the set of maps from the point object), and that all morphisms between the spaces are completely determined by what they do to the underlying set of points. This means that the “spaces” really are just sets with some structure. Now, if the site happens to be $TOP(X)$, then we have a slightly intuition: the “generalized” spaces are something like generalized bundles over $X$, and the “probes” are now sections of such a bundle. A simple example would be an actual sheaf of functions: these are sections of a trivial bundle, since, say, $\mathbb{C}$-valued functions are sections of the bundle $\pi: X \times \mathbb{C} \rightarrow X$. Given a nontrivial bundle $\pi : M \rightarrow X$, there is a sheaf of sections – on each $U$, one gets $F_M(U)$ to be all the one-sided inverses $s : U \rightarrow M$ which are one-sided inverses of $\pi$. For a generic sheaf, we can imagine a sort of “generalized bundle” over $X$. ### Schemes Another example of the fact that sheaves can be seen as spaces is the category of schemes: these are often described as topological spaces which are themselves equipped with a sheaf of rings. “Scheme” is to algebraic geometry what “manifold” is to differential geometry: a kind of space which looks locally like something classical and familiar. Schemes, in some neighborhood of each point, must resemble varieties – i.e. the locus of zeroes of some algebraic function on$\mathbb{k}^n$. For varieties, the rings attached to neighborhoods are rings of algebraic functions on this locus, which will be a quotient of the ring of polynomials. But another way to think of schemes is as concrete sheaves on a site whose objects are varieties and whose morphisms are algebraic maps. This is dual to the other point of view, just as thinking of diffeological spaces as sheaves is dual to a viewpoint in which they’re seen as topological spaces equipped with a notion of “smooth function”. (Some general discussion of this in a talk by Victor Piercey) ### Generalities These two viewpoints (defining the structure of a space by a class of maps into it, or by a class of maps out of it) in principle give different definitions. To move between them, you really need everything to be concrete: the space has an underlying set, the set of probes is a collection of real set-functions. Likewise, for something like a scheme, you’d need the ring for any open set to be a ring of actual set-functions. In this case, one can move between the two descriptions of the space as long as there is a pre-existing concept of the right kind of function on the “probe” spaces. Given a smooth space, say, one can define a sheaf of smooth functions on each open set by taking those whose composites with every probe are smooth. Conversely, given something like a scheme, where the structure sheaf is of function rings on each open subspace (i.e. the sheaf is representable), one can define the probes from varieties to be those which give algebraic functions when composed with every function in these rings. Neither of these will work in general: the two approaches define different categories of spaces (in the smooth context, see Andrew Stacey’s comparison of various categories of smooth spaces, defined either by specifying the smooth maps in, or out, or both). But for very concrete situations, they fit together neatly. The concrete case is therefore nice for getting an intuition for what it means to think of sheaves as spaces. For sheaves which aren’t concrete, morphisms aren’t determined by what they do to the underlying points i.e. the forgetful “underlying set” functor isn’t faithful. Here, we might think of a “generalized space” which looks like two copies of the same topological space: the sheaf gives two different elements of $F(U)$ for each map of underlying sets. We could think of such generalized space as built from sets equipped with extra “stuff” (say, a set consisting of pairs $(x,i) \in X \times \{ blue , green \}$ – so it consists of a “blue” copy of X and a “green” copy of X, but the underlying set functor ignores the colouring. Still, useful as they may be to get a first handle on this concept of sheaf as generalized space, one shouldn’t rely on these intuitions too much: if $\mathcal{T}$ doesn’t even have a “point” object, there is no underlying set functor at all. Eventually, one simply has to get used to the idea of defining a space by the information revealed by probes. In the next post, I’ll talk more about this in the context of John Huerta’s guest lecture, applying this idea to the category of supermanifolds, which can be seen as manifolds built internal to the topos of (pre)sheaves on a site whose objects are called “super-points”. This entry is a by-special-request blog, which Derek Wise invited me to write for the blog associated with the International Loop Quantum Gravity Seminar, and it will appear over there as well. The ILQGS is a long-running regular seminar which runs as a teleconference, with people joining in from various countries, on various topics which are more or less closely related to Loop Quantum Gravity and the interests of people who work on it. The custom is that when someone gives a talk, someone else writes up a description of the talk for the ILQGS blog, and Derek invited me to write up a description of his talk. The audio file of the talk itself is available in .aiff and .wav formats, and the slides are here. The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a few papers (two shorter ones, “Spontaneously broken Lorentz symmetry for Hamiltonian gravity“, “Linking Covariant and Canonical General Relativity via Local Observers“, and a new, longer one called “Lifting General Relativity to Observer Space“). The key idea behind this project is the notion of “observer space”, which is exactly what it sounds like: a space of all observers in a given universe. This is easiest to picture when one has a spacetime – a manifold with a Lorentzian metric, $(M,g)$ – to begin with. Then an observer can be specified by choosing a particular point $(x_0,x_1,x_2,x_3) = \mathbf{x}$ in spacetime, as well as a unit future-directed timelike vector $v$. This vector is a tangent to the observer’s worldline at $\mathbf{x}$. The observer space is therefore a bundle over $M$, the “future unit tangent bundle”. However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying $(M,g)$. The result is a surprising, relatively new physical intuition is that “spacetime” is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same spacetime. This is somewhat related to the relativity of locality, which I’ve blogged about previously. Geometrically, it is similar to the fact that a slicing of spacetime into space and time is not unique, and not respected by the full symmetries of the theory of Relativity, even for flat spacetime (much less for the case of General Relativity). Similarly, we will see a notion of “observer space”, which can sometimes be turned into a bundle over an objective spacetime $M$, but not in all cases. So, how is this described mathematically? In particular, what did I mean up there by saying that spacetime becomes observer-dependent? ### Cartan Geometry The answer uses Cartan geometry, which is a framework for differential geometry that is slightly broader than what is commonly used in physics. Roughly, one can say “Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry”. The more familiar direction of generalization here is the fact that, like Riemannian geometry, Cartan is concerned with manifolds which have local models in terms of simple, “flat” geometries, but which have curvature, and fail to be homogeneous. First let’s remember how Klein geometry works. Klein’s Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role. It describes “homogeneous spaces”, which are geometries in which every point is indistinguishable from every other point. This is expressed by the existence of a transitive action of some Lie group $G$ of all symmetries on an underlying space. Any given point $x$ will be fixed by some symmetries, and not others, so one also has a subgroup $H = Stab(x) \subset G$. This is the “stabilizer subgroup”, consisting of all symmetries which fix $x$. That the space is homogeneous means that for any two points $x,y$, the subgroups $Stab(x)$ and $Stab(y)$ are conjugate (by a symmetry taking $x$ to $y$). Then the homogeneous space, or Klein geometry, associated to $(G,H)$ is, up to isomorphism, just the same as the quotient space $G/H$ of the obvious action of $H$ on $G$. The advantage of this program is that it has a great many examples, but the most relevant ones for now are: • $n$-dimensional Euclidean space. the Euclidean group $ISO(n) = SO(n) \ltimes \mathbb{R}^n$ is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant. It acts transitively on $\mathbb{R}^n$. Any point will be fixed by the group of rotations centred at that point, which is a subgroup of $ISO(n)$ isomorphic to $SO(n)$. Klein’s insight is to reverse this: we may define Euclidean space by $R^n \cong ISO(n)/SO(n)$. • $n$-dimensional Minkowski space. Similarly, we can define this space to be $ISO(n-1,1)/SO(n-1,1)$. The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same. • de Sitter space. As a Klein geometry, this is the quotient $SO(4,1)/SO(3,1)$. That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point. But the global symmetries of de Sitter space are different. Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations $so(4,1)/so(3,1)$ and $iso(3,1)/so(3,1)$ are identical, seen as representations of $SO(3,1)$. It’s natural to identify them with the tangent space at a point. de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in $\mathbb{R}^5$. This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature. • anti-de Sitter space. This is similar, but now the quotient is $SO(3,2)/SO(3,1)$ – in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative. Now, what does it mean to say that a Cartan geometry has a local model? Well, just as a Lorentzian or Riemannian manifold is “locally modelled” by Minkowski or Euclidean space, a Cartan geometry is locally modelled by some Klein geometry. This is best described in terms of a connection on a principal $G$-bundle, and the associated $G/H$-bundle, over some manifold $M$. The crucial bundle in a Riemannian or Lorenztian geometry is the frame bundle: the fibre over each point consists of all the ways to isometrically embed a standard Euclidean or Minkowski space into the tangent space. A connection on this bundle specifies how this embedding should transform as one moves along a path. It’s determined by a 1-form on $M$, valued in the Lie algebra of $G$. Given a parametrized path, one can apply this form to the tangent vector at each point, and get a Lie algebra-valued answer. Integrating along the path, we get a path in the Lie group $G$ (which is independent of the parametrization). This is called a “development” of the path, and by applying the $G$-values to the model space $G/H$, we see that the connection tells us how to move through a copy of $G/H$ as we move along the path. The image this suggests is of “rolling without slipping” – think of the case where the model space is a sphere. The connection describes how the model space “rolls” over the surface of the manifold $M$. Curvature of the connection measures the failure to commute of the processes of rolling in two different directions. A connection with zero curvature describes a space which (locally at least) looks exactly like the model space: picture a sphere rolling against its mirror image. Transporting the sphere-shaped fibre around any closed curve always brings it back to its starting position. Now, curvature is defined in terms of transports of these Klein-geometry fibres. If curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model. This idea, that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics. ### Gravity and Cartan Geometry MacDowell-Mansouri gravity can be understood as a theory in which General Relativity is modelled by a Cartan geometry. Of course, a standard way of presenting GR is in terms of the geometry of a Lorentzian manifold. In the Palatini formalism, the basic fields are a connection $A$ and a vierbein (coframe field) called $e$, with dynamics encoded in the Palatini action, which is the integral over $M$ of $R[\omega] \wedge e \wedge e$, where $R$ is the curvature 2-form for $\omega$. This can be derived from a Cartan geometry, whose model geometry is de Sitter space $SO(4,1)/SO(3,1)$. Then MacDowell-Mansouri gravity gets $\omega$ and $e$ by splitting the Lie algebra as $so(4,1) = so(3,1) \oplus \mathbb{R^4}$. This “breaks the full symmetry” at each point. Then one has a fairly natural action on the $so(4,1)$-connection: $\int_M tr(F_h \wedge \star F_h)$ Here, $F_h$ is the $so(3,1)$ part of the curvature of the big connection. The splitting of the connection means that $F_h = R + e \wedge e$, and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action). So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space. The cosmological constant in GR shows up because a “flat” connection for a Cartan geometry based on de Sitter space will look (if measured by Minkowski space) as if it has constant curvature which is exactly that of the model Klein geometry. The way to think of this is to take the fibre bundle of homogeneous model spaces as a replacement for the tangent bundle to the manifold. The fibre at each point describes the local appearance of spacetime. If empty spacetime is flat, this local model is Minkowski space, $ISO(3,1)/SO(3,1)$, and one can really speak of tangent “vectors”. The tangent homogeneous space is not linear. In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of $Lie(G)$-valued forms can be treated in the same way, regardless of which Klein geometry $G/H$ occurs in the fibres. In particular, General Relativity, formulated in terms of Cartan geometry, always says that, in the absence of matter, the geometry of space is flat, and the cosmological constant is included naturally by the choice of which Klein geometry is the local model of spacetime. ### Observer Space The idea in defining an observer space is to combine two symmetry reductions into one. The reduction from $SO(4,1)$ to $SO(3,1)$ gives de Sitter space, $SO(4,1)/SO(3,1)$ as a model Klein geometry, which reflects the “symmetry breaking” that happens when choosing one particular point in spacetime, or event. Then, the reduction of $SO(3,1)$ to $SO(3)$ similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector). These are the tangent vectors to the worldline of an observer at the chosen point, so $SO(3,1)/SO(3)$ the model Klein geometry, is the space of such possible observers. The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in $SO(3,1)$ translate between observers. So locally, choosing an observer amounts to a splitting of the model spacetime at the point into a product of space and time. If we combine both reductions at once, we get the 7-dimensional Klein geometry $SO(4,1)/SO(3)$. This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers” A general observer space $O$, however, is just a Cartan geometry modelled on $SO(4,1)/SO(3)$. This is a 7-dimensional manifold, equipped with the structure of a Cartan geometry. One class of examples are exactly the future unit tangent bundles to 4-dimensional Lorentzian spacetimes. In these cases, observer space is naturally a contact manifold: that is, it’s an odd-dimensional manifold equipped with a 1-form $\alpha$, the contact form, which is such that the top-dimensional form $\alpha \wedge d \alpha \wedge \dots \wedge d \alpha$ is nowhere zero. This is the odd-dimensional analog of a symplectic manifold. Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane. In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold $M$. Now, Minkowski space has a slicing into space and time – in fact, one for each observer, who defines the time direction, but the time coordinate does not transform in any meaningful way under the symmetries of the theory, and different observers will choose different ones. In just the same way, the homogeneous model of observer space can naturally be written as a bundle $SO(4,1)/SO(3) \rightarrow SO(4,1)/SO(3,1)$. But a general observer space $O$ may or may not be a bundle over an ordinary spacetime manifold, $O \rightarrow M$. Every Cartan geometry $M$ gives rise to an observer space $O$ as the bundle of future-directed timelike vectors, but not every Cartan geometry $O$ is of this form, in any natural way. Indeed, without a further condition, we can’t even reconstruct observer space as such a bundle in an open neighborhood of a given observer. This may be intuitively surprising: it gives a perfectly concrete geometric model in which “spacetime” is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between “space” and “time” in General Relativity. It may be impossible, that is, to determine objectively whether two observers are located at the same base event or not. This is a kind of “Relativity of Locality” which is geometrically much like the by-now more familiar Relativity of Simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree. The coincident observers according to a given observer are those reached by a good class of geodesics in $O$ moving only in directions that observer sees as boosts. When one can reconstruct $O \rightarrow M$, two observers will agree whether or not they are coincident. This extra condition which makes this possible is an integrability constraint on the action of the Lie algebra $H$ (in our main example, $H = SO(3,1)$) on the observer space $O$. In this case, the fibres of the bundle are the orbits of this action, and we have the familiar world of Relativity, where simultaneity may be relative, but locality is absolute. ### Lifting Gravity to Observer Space Apart from describing this model of relative spacetime, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) GR locally near each point in such an observer space. The goal is to make a link between covariant and canonical quantization of gravity. Covariant quantization treats the geometry of spacetime all at once, by means of a Lagrangian action functional. This is mathematically appealing, since it respects the symmetry of General Relativity, namely its diffeomorphism-invariance. On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one gets a Hilbert space by quantizing the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution. This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer. The point of observer space is that we consider all these choices at once. Describing GR in $O$ is both covariant, and based on (local) choices of time direction. This is easiest to describe in the case of a bundle $O \rightarrow M$. Then a “field of observers” to be a section of the bundle: a choice, at each base event in $M$, of an observer based at that event. A field of observers may or may not correspond to a particular decomposition of spacetime into space evolving in time, but locally, at each point in $O$, it always looks like one. The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer. In this case, a Cartan connection on observer space is described by to a $Lie(SO(4,1))$-valued form. This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation. The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of $SO(4,1)$ to $SO(3,1)$, which symmetry breaking entails choosing a point); and second, between space and time (i.e. the reduction of $SO(3,1)$ to $SO(3)$, which symmetry breaking entails choosing a time direction). This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift GR to a theory on an observer space $O$. This amount to describing fields on $O$ and an action functional, so that the splitting of the fields gives back the usual fields of GR on spacetime, and the action gives back the usual action. This part of the project is still under development, but this lifting has been described. In the case when there is no “objective” spacetime, the result includes some surprising new fields which it’s not clear how to deal with, but when there is an objective spacetime, the resulting theory looks just like GR. Since the last post, I’ve been busily attending some conferences, as well as moving to my new job at the University of Hamburg, in the Graduiertenkolleg 1670, “Mathematics Inspired by String Theory and Quantum Field Theory”. The week before I started, I was already here in Hamburg, at the conference they were organizing “New Perspectives in Topological Quantum Field Theory“. But since I last posted, I was also at the 20th Oporto Meeting on Geometry, Topology, and Physics, as well as the third Higher Structures in China workshop, at Jilin University in Changchun. Right now, I’d like to say a few things about some of the highlights of that workshop. Higher Structures in China III So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv. In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification. Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups. That is still under development, but I was invited to give a couple of talks on the subject at the workshop. It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East. (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there). It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around. So that was a sort of holiday, with 11 other mathematicians. Here is me with Dany Majard, in a national park along the way to the mountains: Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China): And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater. The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China: Well, that was fun! Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University. So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another. We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students. ### Categorified Algebra This is a sort of broad term, but certainly would include my own talk. The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category. The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors. This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples. I’ve discussed this general subject in recent posts, so I won’t elaborate too much. The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary. Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case $U_q(sl_2)$, somewhat along the same lines as in his introductory paper on the subject. The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups. The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots. One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum $sl_2$ on categories. This starting point is nice, because it can work by just mimicking the construction of $sl_2$ and $U_q(sl_2)$ representations in terms of weight spaces: one gets categories $V_{-N}, \dots, V_N$ which correspond to the “weight spaces” (usually just vector spaces), and the $E$ and $F$ operators give functors between them, and so forth. Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy. One such example involves flag varieties. A flag, as one might eventually guess from the name, is a nested collection of subspaces in some $n$-dimensional space. A simple example is the Grassmannian $Gr(1,V)$, which is the space of all 1-dimensional subspaces of $V$ (i.e. the projective space $P(V)$), which is of course an algebraic variety. Likewise, $Gr(k,V)$, the space of all $k$-dimensional subspaces of $V$ is a variety. The flag variety $Fl(k,k+1,V)$ consists of all pairs $W_k \subset W_{k+1}$, of a $k$-dimensional subspace of $V$, inside a $(k+1)$-dimensional subspace (the case $k=2$ calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole). This collection is again a variety. One can go all the way up to the variety of “complete flags”, $Fl(1,2,\dots,n,V)$ (where $V$ is $n$-dimenisonal), any point of which picks out a subspace of each dimension, each inside the next. The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form $Fl(k,k+1,V)$ as relating the Grassmanians: there are projections $Fl(k,k+1,V) \rightarrow Gr(k,V)$ and $Fl(k,k+1,V) \rightarrow Gr(k+1,V)$, which act by just ignoring one or the other of the two subspaces of a flag. This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces. So one gets a sequence $H_0, H_1, \dots, H_n$, and maps between the adjacent ones. This becomes a representation of the Lie algebra. Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories. So one gets a categorified representation. Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology. The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra. The two look the same over a field of characteristic 2, but otherwise different. The idea is then that there should be “odd” versions of various structures that show up in the categorifications of $U_q(sl_2)$ (and other algebras) mentioned above. One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators. This is an algebra which can be seen to act on the space of polynomials in $n$ commuting variables, $\mathbb{C}[x_1,\dots,x_n]$, generated by the multiplication operators $x_i$, and the “divided difference operators” based on the swapping of two adjacent variables. The Hecke algebra is defined in terms of “swap” generators, which satisfy some $q$-deformed variation of the relations that define the symmetric group (and hence its group algebra). The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero. The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra. The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the $x_i$ are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation). This sort of “oddification” is then supposed to happen all over. The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties. Then the odd Nil Hecke algebra acts on that, analogously to the even case above. Marco Mackaay gave a couple of talks about the $sl_3$ web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer. This is the analog of the above, for $U_q(sl_3)$, describing a diagram calculus which accounts for representations of the quantum group. The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these. When categorifying, one gets a calculus of “foams” between such diagrams. Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well. ### Algebraic Structures Some of the talks were about structures in algebra in a more conventional sense. Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras. The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product. That is, the associative product of two elements gives a power series in some parameter $h$ (which one should think of as something like Planck’s constant in a quantum setting). The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term. This gives a Poisson bracket (or rather, the commutator of the associative product does). In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits. The construction uses words in the generating reflections for the Weyl group of a Lie group $G$. Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on. Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions. In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc. Taken as a big pile of definitions of complicated structures, this seems like a terrible mess. The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold. The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations. The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator. The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations. Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures. Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible. The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions. Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories. Double categories have ordinary categories (and hence monoids and sets) as degenerate cases. Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids. There is also the special degenerate case of a double monoid (and the extra-special case of a double group). (The slides have nice pictures showing how they’re all degenerate cases). Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product” $H \bowtie K$ of two groups $H$ and $K$ (each of which has to act on the other for this to make sense). He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition: $Poinc = (SO(3) \bowtie (SO(1; 1) \bowtie N)) \ltimes \mathbb{R}_4$ ($N$ is certain group of matrices). So there’s a unique double group which corresponds to it – it has squares labelled by $\mathbb{R}_4$, and the horizontial and vertical morphisms by elements of $SO(3)$ and $N$ respectively. Dany finished by explaining that there are higher-dimensional analogs of all this – $n$-tuple categories can be defined recursively by internalization (“internal categories in $(n-1)$-tuple-Cat”). There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of $n$-tuple groups. The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an $n$-fold iterated bicrossed product of groups. Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators). FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series. (So, in particular they have an underlying ring, for this to make sense). In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras. Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper. ### Physics I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature. Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program. There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do. Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories. For instance, the usual cohomology of a space $X$ is a quotient of the space of closed forms (so the $k^{th}$ cohomology, $H^{k}(X) = \Omega^{k}$, is a quotient of the space of closed $k$-forms – the quotient being that forms differing by a coboundary are considered the same). There’s a similar construction for the $K$-theory $K(X)$, which can be modelled as a quotient of the space of vector bundles over $X$. Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around. The first two of these examples are known to be related to certain supersymmetric topological quantum field theories. Now, a TFT is a functor into some kind of vector spaces from a category of $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms $Z : d-Bord \rightarrow Vect$ Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime. A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension. A normal smooth manifold is a ringed space $(M,\mathcal{O})$, since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely). Supersymmetric theories work with manifolds which change this sheaf – so a $d|\delta$-dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions $\theta_i$, the “fermionic dimensions”: $\mathcal{O}(U) = C^{\infty}(U) \otimes \bigwedge^{\ast}[\theta_1,\dots,\theta_{\delta}]$ Then a supersymmetric TFT is a functor: $E : (d|\delta)-Bord \rightarrow STV$ (where $STV$ is the category of supersymmetric topological vector spaces – defined similarly). The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories. Ordinary cohomology corresponds then to $0|1$-dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and $K$-theory to a $1|1$-dimensional extended TFT. The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a $2_1$-dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s. Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers. This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds. I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example. Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space. The point is that, if points in a space are objects and paths are morphisms, then a loop space $L(X)$ shifts things by one categorical level: its points are loops in $X$, and its paths are therefore certain 2-morphisms of $X$. In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space. Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space. The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space. Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of $X$. The general notion of n-plectic geometry, where the usual symplectic geometry is the case $n=1$, involves a $(n+1)$-form analogous to the usual symplectic form. Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs. But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold. An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group $G$ together with a 3-form naturally constructed from its Maurer-Cartan form. At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience! (Note: WordPress seems to be having some intermittent technical problem parsing my math markup in this post, so please bear with me until it, hopefully, goes away…) As August is the month in which Portugal goes on vacation, and we had several family visitors toward the end of the summer, I haven’t posted in a while, but the term has now started up at IST, and seminars are underway, so there should be some interesting stuff coming up to talk about. New Blog First, I’ll point out that that Derek Wise has started a new blog, called simply “Simplicity“, which is (I imagine) what it aims to contain: things which seem complex explained so as to reveal their simplicity. Unless I’m reading too much into the title. As of this writing, he’s posted only one entry, but a lengthy one that gives a nice explanation of a program for categorified Klein geometries which he’s been thinking a bunch about. Klein’s program for describing the geometry of homogeneous spaces (such as spherical, Euclidean, and hyperbolic spaces with constant curvature, for example) was developed at Erlangen, and goes by the name “The Erlangen Program”. Since Derek is now doing a postdoc at Erlangen, and this is supposed to be a categorification of Klein’s approach, he’s referred to it the “2-Erlangen Program”. There’s more discussion about it in a (somewhat) recent post by John Baez at the n-Category Cafe. Both of them note the recent draft paper they did relating a higher gauge theory based on the Poincare 2-group to a theory known as teleparallel gravity. I don’t know this theory so well, except that it’s some almost-equivalent way of formulating General Relativity I’ll refer you to Derek’s own post for full details of what’s going on in this approach, but the basic motivation isn’t too hard to set out. The Erlangen program takes the view that a homogeneous space is a space $X$ (let’s say we mean by this a topological space) which “looks the same everywhere”. More precisely, there’s a group action by some $G$, which we understand to be “symmetries” of the space, which is transitive. Since every point is taken to every other point by some symmetry, the space is “homogeneous”. Some symmetries leave certain points $x \in X$ where they are – they form the stabilizer subgroup $H = Stab(x)$. When the space is homogeneous, it is isomorphic to the coset space, $X \cong G / H$. So Klein’s idea is to say that any time you have a Lie group $G$ and a closed subgroup $H < G$, this quotient will be called a “homogeneous space”. A familiar example would be Euclidean space, $\mathbb{R}^n \cong E(n) / O(n)$, where $E$ is the Euclidean group and $O$ is the orthogonal group, but there are plenty of others. This example indicates what Cartan geometry is all about, though – this is the next natural step after Klein geometry (Edit: Derek’s blog now has a visual explanation of Cartan geometry, a.k.a. “generalized hamsterology”, new since I originally posted this). We can say that Cartan is to Klein as Riemann is to Euclid. (Or that Cartan is to Riemann as Klein is to Euclid – or if you want to get maybe too-precisely metaphorical, Cartan is the pushout of Klein and Riemann over Euclid). The point is that Riemannian geometry studies manifolds – spaces which are not homogeneous, but look like Euclidean space locally. Cartan geometry studies spaces which aren’t homogeneous, but can be locally modelled by Klein geometries. Now, a Riemannian geometry is essentially a manifold with a metric, describing how it locally looks like Euclidean space. An equivalent way to talk about it is a manifold with a bundle of Euclidean spaces (the tangent spaces) with a connection (the Levi-Civita connection associated to the metric). A Cartan geometry can likewise be described as a $G$-bundle with fibre $X$ with a connection Then the point of the “2-Erlangen program” is to develop similar geometric machinery for 2-groups (a.k.a. categorical groups). This is, as usual, a bit more complicated since actions of 2-groups are trickier than group-actions. In their paper, though, the point is to look at spaces which are locally modelled by some sort of 2-Klein geometry which derives from the Poincare 2-group. By analogy with Cartan geometry, one can talk about such Poincare 2-group connections on a space – that is, some kind of “higher gauge theory”. This is the sort of framework where John and Derek’s draft paper formulates teleparallel gravity. It turns out that the 2-group connection ends up looking like a regular connection with torsion, and this plays a role in that theory. Their draft will give you a lot more detail. Talk on Manifold Calculus On a different note, one of the first talks I went to so far this semester was one by Pedro Brito about “Manifold Calculus and Operads” (though he ran out of time in the seminar before getting to talk about the connection to operads). This was about motivating and introducing the Goodwillie Calculus for functors between categories of spaces. (There are various references on this, but see for instance these notes by Hal Sadofsky). In some sense this is a generalization of calculus from functions to functors, and one of the main results Goodwillie introduced with this subject, is a functorial analog of Taylor’s theorem. I’d seen some of this before, but this talk was a nice and accessible intro to the topic. So the starting point for this “Manifold Calculus” is that we’d like to study functors from spaces to spaces (in fact this all applies to spectra, which are more general, but Pedro Brito’s talk was focused on spaces). The sort of thing we’re talking about is a functor which, given a space $M$, gives a moduli space of some sort of geometric structures we can put on $M$, or of mappings from $M$. The main motivating example he gave was the functor $Imm(-,N) : [Spaces] \rightarrow [Spaces]$ for some fixed manifold $N$. Given a manifold $M$, this gives the mapping space of all immersions of $M$ into $N$. (Recalling some terminology: immersions are maps of manifolds where the differential is nondegenerate – the induced map of tangent spaces is everywhere injective, meaning essentially that there are no points, cusps, or kinks in the image, but there might be self-intersections. Embeddings are, in addition, local homeomorphisms.) Studying this functor $Imm(-,N)$ means, among other things, looking at the various spaces $Imm(M,N)$ of immersions of each $M$ into $N$. We might first ask: can $M$ be immersed in $N$ at all – in other words, is $\pi_0(Imm(M,N))$ nonempty? So, for example, the Whitney Embedding Theorem says that if $dim(N)$ is at least $2 dim(M)$, then there is an embedding of $M$ into $N$ (which is therefore also an immersion). In more detail, we might want to know what $\pi_0(Imm(M,N))$ is, which tells how many connected components of immersions there are: in other words, distinct classes of immersions which can’t be deformed into one another by a family of immersions. Or, indeed, we might ask about all the homotopy groups of $Imm(M,N)$, not just the zeroth: what’s the homotopy type of $Imm(M,N)$? (Once we have a handle on this, we would then want to vary $M$). It turns out this question is manageable, party due to a theorem of Smale and Hirsch, which is a generalization of Gromov’s h-principle – the original principle applies to solutions of certain kinds of PDE’s, saying that any solution can be deformed to a holomorphic one, so if you want to study the space of solutions up to homotopy, you may as well just study the holomorphic solutions. The Smale-Hirsch theorem likewise gives a homotopy equivalence of two spaces, one of which is $Imm(M,N)$. The other is the space of “formal immersions”, called $Imm^f(M,N)$. It consists of all $(f,F)$, where $f : M \rightarrow N$ is smooth, and $F : TM \rightarrow TN$ is a map of tangent spaces which restricts to $f$, and is injective. These are “formally” like immersions, and indeed $Imm(M,N)$ has an inclusion into $Imm^f(M,N)$, which happens to be a homotopy equivalence: it induces isomorphisms of all the homotopy groups. These come from homotopies taking each “formal immersion” to some actual immersion. So we’ve approximated $Imm(-,N)$, up to homotopy, by $Imm^f(-,N)$. (This “homotopy” of functors makes sense because we’re talking about an enriched functor – the source and target categories are enriched in spaces, where the concepts of homotopy theory are all available). We still haven’t got to manifold calculus, but it will be all about approximating one functor by another – or rather, by a chain of functors which are supposed to be like the Taylor series for a function. The way to get this series has to do with sheafification, so first it’s handy to re-describe what the Smale-Hirsch theorem says in terms of sheaves. This means we want to talk about some category of spaces with a Grothendieck topology. So lets let $\mathcal{E}$ be the category whose objects are $d$-dimensional manifolds and whose morphisms are embeddings (which, of course, are necessarily codimension 0). Now, the point here is that if $f : M \rightarrow M'$ is an embedding in $\mathcal{E}$, and $M'$ has an immersion into $N$, this induces an immersion of $M$ into $N$. This amounst to saying $Imm(-,N)$ is a contravariant functor: $Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces]$ That makes $Imm(-,N)$ a presheaf. What the Smale-Hirsch theorem tells us is that this presheaf is a homotopy sheaf – but to understand that, we need a few things first. First, what’s a homotopy sheaf? Well, the condition for a sheaf says that if we have an open cover of $M$, then So to say how $Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces]$ is a homotopy sheaf, we have to give $\mathcal{E}$ a topology, which means defining a “cover”, which we do in the obvious way – a cover is a collection of morphisms $f_i : U_i \rightarrow M$ such that the union of all the images $\cup f_i(U_i)$ is just $M$. The topology where this is the definition of a cover can be called $J_1$, because it has the property that given any open cover and choice of 1 point in $M$, that point will be in some $U_i$ of the cover. This is part of a family of topologies, where $J_k$ only allows those covers with the property that given any choice of $k$ points in $M$, some open set of the cover contains them all. These conditions, clearly, get increasingly restrictive, so we have a sequence of inclusions (a “filtration”): $J_1 \leftarrow J_2 \leftarrow J_3 \leftarrow \dots$ Now, with respect to any given one of these topologies $J_k$, we have the usual situation relating sheaves and presheaves. Sheaves are defined relative to a given topology (i.e. a notion of cover). A presheaf on $\mathcal{E}$ is just a contravariant functor from $\mathcal{E}$ (in this case valued in spaces); a sheaf is one which satisfies a descent condition (I’ve discussed this before, for instance here, when I was running the Stacks Seminar at UWO). The point of a descent condition, for a given topology is that if we can take the values of a functor $F$ “locally” – on the various objects of a cover for $M$ – and “glue” them to find the value for $M$ itself. In particular, given a cover for $M \in \mathcal{E}$, and a cover, there’s a diagram consisting of the inclusions of all the double-overlaps of sets in the cover into the original sets. Then the descent condition for sheaves of spaces is that The general fact is that there’s a reflective inclusion of sheaves into presheaves (see some discussion about reflective inclusions, also in an earlier post). Any sheaf is a contravariant functor – this is the inclusion of $Sh( \mathcal{E} )$ into$latex PSh( \mathcal{E} )$. The reflection has a left adjoint, sheafification, which takes any presheaf in $PSh( \mathcal{E} )$ to a sheaf which is the “best approximation” to it. It’s the fact this is an adjoint which makes the inclusion “reflective”, and provides the sense in which the sheafification is an approximation to the original functor. The way sheafification works can be worked out from the fact that it’s an adjoint to the inclusion, but it also has a fairly concrete description. Given any one of the topologies $J_k$, we have a whole collection of special diagrams, such as: $U_i \leftarrow U_{ij} \rightarrow U_j$ (using the usual notation where $U_{ij} = U_i \cap U_j$ is the intersection of two sets in a cover, and the maps here are the inclusions of that intersection). This and the various other diagrams involving these inclusions are special, given the topology $J_k$. The descent condition for a sheaf $F$ says that if we take the image of this diagram: $F(U_i) \rightarrow F(U_{ij}) \leftarrow F(U_j)$ then we can “glue together” the objects $F(U_i)$ and $F(U_j)$ on the overlap to get one on the union. That is, $F$ is a sheaf if $F(U_i \cup U_j)$ is a colimit of the diagram above (intuitively, by “gluing on the overlap”). In a presheaf, it would come equipped with some maps into the $F(U_i)$ and $F(U_j)$: in a sheaf, this object and the maps satisfy some universal property. Sheafification takes a presheaf $F$ to a sheaf $F^{(k)}$ which does this, essentially by taking all these colimits. More accurately, since these sheaves are valued in spaces, what we really want are homotopy sheaves, where we can replace “colimit” with “homotopy colimit” in the above – which satisfies a universal property only up to homotopy, and which has a slightly weaker notion of “gluing”. This (homotopy) sheaf is called $F^{(k)}$ because it depends on the topology $J_k$ which we were using to get the class of special diagrams. One way to think about $F^{(k)}$ is that we take the restriction to manifolds which are made by pasting together at most $k$ open balls. Then, knowing only this part of the functor $F$, we extend it back to all manifolds by a Kan extension (this is the technical sense in which it’s a “best approximation”). Now the point of all this is that we’re building a tower of functors that are “approximately” like $F$, agreeing on ever-more-complicated manifolds, which in our motivating example is $F = Imm(-,N)$. Whichever functor we use, we get a tower of functors connected by natural transformations: $F^{(1)} \leftarrow F^{(2)} \leftarrow F^{(3)} \leftarrow \dots$ This happens because we had that chain of inclusions of the topologies $J_k$. Now the idea is that if we start with a reasonably nice functor (like $F = Imm(-,N)$ for example), then $F$ is just the limit of this diagram. That is, it’s the universal thing $F$ which has a map into each $F^{(k)}$ commuting with all these connecting maps in the tower. The tower of approximations – along with its limit (as a diagram in the category of functors) – is what Goodwillie called the “Taylor tower” for $F$. Then we say the functor $F$ is analytic if it’s just (up to homotopy!) the limit of this tower. By analogy, think of an inclusion of a vector space $V$ with inner product into another such space $W$ which has higher dimension. Then there’s an orthogonal projection onto the smaller space, which is an adjoint (as a map of inner product spaces) to the inclusion – so these are like our reflective inclusions. So the smaller space can “reflect” the bigger one, while not being able to capture anything in the orthogonal complement. Now suppose we have a tower of inclusions $V \leftarrow V' \leftarrow V'' \dots$, where each space is of higher dimension, such that each of the $V$ is included into $W$ in a way that agrees with their maps to each other. Then given a vector $w \in W$, we can take a sequence of approximations $(v,v',v'',\dots)$ in the $V$ spaces. If $w$ was “nice” to begin with, this series of approximations will eventually at least converge to it – but it may be that our tower of $V$ spaces doesn’t let us approximate every $w$ in this way. That’s precisely what one does in calculus with Taylor series: we have a big vector space $W$ of smooth functions, and a tower of spaces we use to approximate. These are polynomial functions of different degrees: first linear, then quadratic, and so forth. The approximations to a function $f$ are orthogonal projections onto these smaller spaces. The sequence of approximations, or rather its limit (as a sequence in the inner product space $W$), is just what we mean by a “Taylor series for $f$“. If $f$ is analytic in the first place, then this sequence will converge to it. The same sort of phenomenon is happening with the Goodwillie calculus for functors: our tower of sheafifications of some functor $F$ are just “projections” onto smaller categories (of sheaves) inside the category of all contravariant functors. (Actually, “reflections”, via the reflective inclusions of the sheaf categories for each of the topologies $J_k$). The Taylor Tower for this functor is just like the Taylor series approximating a function. Indeed, this analogy is fairly close, since the topologies $J_k$ will give approximations of $F$ which are in some sense based on $k$ points (so-called $k$-excisive functors, which in our terminology here are sheaves in these topologies). Likewise, a degree-$k$ polynomial approximation approximates a smooth function, in general in a way that can be made to agree at $k$ points. Finally, I’ll point out that I mentioned that the Goodwillie calculus is actually more general than this, and applies not only to spaces but to spectra. The point is that the functor $Imm(-,N)$ defines a kind of generalized cohomology theory – the cohomology groups for $M$ are the $\pi_i(Imm(M,N))$. So the point is, functors satisfying the axioms of a generalized cohomology theory are represented by spectra, whereas $N$ here is a special case that happens to be a space. Lots of geometric problems can be thought of as classified by this sort of functor – if $N = BG$, the classifying space of a group, and we drop the requirement that the map be an immersion, then we’re looking at the functor that gives the moduli space of $G$-connections on each $M$. The point is that the Goodwillie calculus gives a sense in which we can understand such functors by simpler approximations to them. So Dan Christensen, who used to be my supervisor while I was a postdoc at the University of Western Ontario, came to Lisbon last week and gave a talk about a topic I remember hearing about while I was there. This is the category $Diff$ of diffeological spaces as a setting for homotopy theory. Just to make things scan more nicely, I’m going to say “smooth space” for “diffeological space” here, although this term is in fact ambiguous (see Andrew Stacey’s “Comparative Smootheology” for lots of details about options). There’s a lot of information about $Diff$ in Patrick Iglesias-Zimmour’s draft-of-a-book. Motivation The point of the category $Diff$, initially, is that it extends the category of manifolds while having some nicer properties. Thus, while all manifolds are smooth spaces, there are others, which allow $Diff$ to be closed under various operations. These would include taking limits and colimits: for instance, any subset of a smooth space becomes a smooth space, and any quotient of a smooth space by an equivalence relation is a smooth space. Then too, $Diff$ has exponentials (that is, if $A$ and $B$ are smooth spaces, so is $A^B = Hom(B,A)$). So, for instance, this is a good context for constructing loop spaces: a manifold $M$ is a smooth space, and so is its loop space $LM = M^{S^1} = Hom(S^1,M)$, the space of all maps of the circle into $M$. This becomes important for talking about things like higher cohomology, gerbes, etc. When starting with the category of manifolds, doing this requires you to go off and define infinite dimensional manifolds before $LM$ can even be defined. Likewise, the irrational torus is hard to talk about as a manifold: you take a torus, thought of as $\mathbb{R}^2 / \mathbb{Z}^2$. Then take a direction in $\mathbb{R}^2$ with irrational slope, and identify any two points which are translates of each other in $\mathbb{R}^2$ along the direction of this line. The orbit of any point is then dense in the torus, so this is a very nasty space, certainly not a manifold. But it’s a perfectly good smooth space. Well, these examples motivate the kinds of things these nice categorical properties allow us to do, but $Diff$ wouldn’t deserve to be called a category of “smooth spaces” (Souriau’s original name for them) if they didn’t allow a notion of smooth maps, which is the basis for most of what we do with manifolds: smooth paths, derivatives of curves, vector fields, differential forms, smooth cohomology, smooth bundles, and the rest of the apparatus of differential geometry. As with manifolds, this notion of smooth map ought to get along with the usual notion for $\mathbb{R}^n$ in some sense. Smooth Spaces Thus, a smooth (i.e. diffeological) space consists of: • A set $X$ (of “points”) • A set $\{ f : U \rightarrow X \}$ (of “plots”) for every n and open $U \subset \mathbb{R}^n$ such that: 1. All constant maps are plots 2. If $f: U \rightarrow X$ is a plot, and $g : V \rightarrow U$ is a smooth map, $f \circ g : V \rightarrow X$ is a plot 3. If $\{ g_i : U_i \rightarrow U\}$ is an open cover of $U$, and $f : U \rightarrow X$ is a map, whose restrictions $f \circ g_i : U_i \rightarrow X$ are all plots, so is $f$ A smooth map between smooth spaces is one that gets along with all this structure (i.e. the composite with every plot is also a plot). These conditions mean that smooth maps agree with the usual notion in $\mathbb{R}^n$, and we can glue together smooth spaces to produce new ones. A manifold becomes a smooth space by taking all the usual smooth maps to be plots: it’s a full subcategory (we introduce new objects which aren’t manifolds, but no new morphisms between manifolds). A choice of a set of plots for some space $X$ is a “diffeology”: there can, of course, be many different diffeologies on a given space. So, in particular, diffeologies can encode a little more than the charts of a manifold. Just for one example, a diffeology can have “stop signs”, as Dan put it – points with the property that any smooth map from $I= [0,1]$ which passes through them must stop at that point (have derivative zero – or higher derivatives, if you like). Along the same lines, there’s a nonstandard diffeology on $I$ itself with the property that any smooth map from this $I$ into a manifold $M$ must have all derivatives zero at the endpoints. This is a better object for defining smooth fundamental groups: you can concatenate these paths at will and they’re guaranteed to be smooth. As a Quasitopos An important fact about these smooth spaces is that they are concrete sheaves (i.e. sheaves with underlying sets) on the concrete site (i.e. a Grothendieck site where objects have underlying sets) whose objects are the $U \subset \mathbb{R}^n$. This implies many nice things about the category $Diff$. One is that it’s a quasitopos. This is almost the same as a topos (in particular, it has limits, colimits, etc. as described above), but where a topos has a “subobject classifier”, a quasitopos has a weak subobject classifier (which, perhaps confusingly, is “weak” because it only classifies the strong subobjects). So remember that a subobject classifier is an object with a map $t : 1 \rightarrow \Omega$ from the terminal object, so that any monomorphism (subobject) $A \rightarrow X$ is the pullback of $t$ along some map $X \rightarrow \Omega$ (the classifying map). In the topos of sets, this is just the inclusion of a one-element set $\{\star\}$ into a two-element set $\{T,F\}$: the classifying map for a subset $A \subset X$ sends everything in $A$ (i.e. in the image of the inclusion map) to $T = Im(t)$, and everything else to $F$. (That is, it’s the characteristic function.) So pulling back $T$ Any topos has one of these – in particular the topos of sheaves on the diffeological site has one. But $Diff$ consists of the concrete sheaves, not all sheaves. The subobject classifier of the topos won’t be concrete – but it does have a “concretification”, which turns out to be the weak subobject classifier. The subobjects of a smooth space $X$ which it classifies (i.e. for which there’s a classifying map as above) are exactly the subsets $A \subset X$ equipped with the subspace diffeology. (Which is defined in the obvious way: the plots are the plots of $X$ which land in $A$). We’ll come back to this quasitopos shortly. The main point is that Dan and his graduate student, Enxin Wu, have been trying to define a different kind of structure on $Diff$. We know it’s good for doing differential geometry. The hope is that it’s also good for doing homotopy theory. As a Model Category The basic idea here is pretty well supported: naively, one can do a lot of the things done in homotopy theory in $Diff$: to start with, one can define the “smooth homotopy groups” $\pi_n^s(X;x_0)$ of a pointed space. It’s a theorem by Dan and Enxin that several possible ways of doing this are equivalent. But, for example, Iglesias-Zimmour defines them inductively, so that $\pi_0^s(X)$ is the set of path-components of $X$, and $\pi_k^s(X) = \pi_{k-1}^s(LX)$ is defined recursively using loop spaces, mentioned above. The point is that this all works in $Diff$ much as for topological spaces. In particular, there are analogs for the $\pi_k^s$ for standard theorems like the long exact sequence of homotopy groups for a bundle. Of course, you have to define “bundle” in $Diff$ – it’s a smooth surjective map $X \rightarrow Y$, but saying a diffeological bundle is “locally trivial” doesn’t mean “over open neighborhoods”, but “under pullback along any plot”. (Either of these converts a bundle over a whole space into a bundle over part of $\mathbb{R}^n$, where things are easy to define). Less naively, the kind of category where homotopy theory works is a model category (see also here). So the project Dan and Enxin have been working on is to give $Diff$ this sort of structure. While there are technicalities behind those links, the essential point is that this means you have a closed category (i.e. with all limits and colimits, which $Diff$ does), on which you’ve defined three classes of morphisms: fibrations, cofibrations, and weak equivalences. These are supposed to abstract the properties of maps in the homotopy theory of topological spaces – in that case weak equivalences being maps that induce isomorphisms of homotopy groups, the other two being defined by having some lifting properties (i.e. you can lift a homotopy, such as a path, along a fibration). So to abstract the situation in $Top$, these classes have to satisfy some axioms (including an abstract form of the lifting properties). There are slightly different formulations, but for instance, the “2 of 3″ axiom says that if two of $f$, latex$g$and $f \circ g$ are weak equivalences, so is the third. Or, again, there should be a factorization for any morphism into a fibration and an acyclic cofibration (i.e. one which is also a weak equivalence), and also vice versa (that is, moving the adjective “acyclic” to the fibration). Defining some classes of maps isn’t hard, but it tends to be that proving they satisfy all the axioms IS hard. Supposing you could do it, though, you have things like the homotopy category (where you formally allow all weak equivalences to have inverses), derived functors(which come from a situation where homotopy theory is “modelled” by categories of chain complexes), and various other fairly powerful tools. Doing this in $Diff$ would make it possible to use these things in a setting that supports differential geometry. In particular, you’d have a lot of high-powered machinery that you could apply to prove things about manifolds, even though it doesn’t work in the category $Man$ itself – only in the larger setting $Diff$. Dan and Enxin are still working on nailing down some of the proofs, but it appears to be working. Their strategy is based on the principle that, for purposes of homotopy, topological spaces act like simplicial complexes. So they define an affine “simplex”, $\mathbb{A}^n = \{ (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} | \sum x_i = 1 \}$. These aren’t literally simplexes: they’re affine planes, which we understand as smooth spaces – with the subspace diffeology from $\mathbb{R}^{n+1}$. But they behave like simplexes: there are face and degeneracy maps for them, and the like. They form a “cosimplicial object”, which we can think of as a functor $\Delta \rightarrow Diff$, where $\Delta$ is the simplex category). Then the point is one can look at, for a smooth space $X$, the smooth singular simplicial set $S(X)$: it’s a simplicial set where the sets are sets of smooth maps from the affine simplex into $X$. Likewise, for a simplicial set $S$, there’s a smooth space, the “geometric realization” $|S|$. These give two functors $|\cdot |$ and $S$, which are adjoints ($| \cdot |$ is the left adjoint). And then, weak equivalences and fibrations being defined in simplicial sets (w.e. are homotopy equivalences of the realization in $Top$, and fibrations are “Kan fibrations”), you can just pull the definition back to $Diff$: a smooth map is a w.e. if its image under $S$ is one. The cofibrations get indirectly defined via the lifting properties they need to have relative to the other two classes. So it’s still not completely settled that this definition actually gives a model category structure, but it’s pretty close. Certainly, some things are known. For instance, Enxin Wu showed that if you have a fibrant object $X$ (i.e. one where the unique map to the terminal object is a fibration – these are generally the “good” objects to define homotopy groups on), then the smooth homotopy groups agree with the simplicial ones for $S(X)$. This implies that for these objects, the weak equivalences are exactly the smooth maps that give isomorphisms for homotopy groups. And so forth. But notice that even some fairly nice objects aren’t fibrant: two lines glued together at a point isn’t, for instance. There are various further results. One, a consquences of a result Enxin proved, is that all manifolds are fibrant objects, where these nice properties apply. It’s interesting that this comes from the fact that, in $Diff$, every (connected) manifold is a homogeneous space. These are quotients of smooth groups, $G/H$ – the space is a space of cosets, and $H$ is understood to be the stabilizer of the point. Usually one thinks of homogenous spaces as fairly rigid things: the Euclidean plane, say, where $G$ is the whole Euclidean group, and $H$ the rotations; or a sphere, where $G$ is all n-dimensional rotations, and $H$ the ones that fix some point on the sphere. (Actually, this gives a projective plane, since opposite points on the sphere get identified. But you get the idea). But that’s for Lie groups. The point is that $G = Diff(M,M)$, the space of diffeomorphisms from $M$ to itself, is a perfectly good smooth group. Then the subgroup $H$ of diffeomorphisms that fix any point is a fine smooth subgroup, and $G/H$ is a homogeneous space in $Diff$. But that’s just $M$, with $G$ acting transitively on it – any point can be taken anywhere on $M$. Cohesive Infinity-Toposes One further thing I’d mention here is related to a related but more abstract approach to the question of how to incorporate homotopy-theoretic tools with a setting that supports differential geometry. This is the notion of a cohesive topos, and more generally of a cohesive infinity-topos. Urs Schreiber has advocated for this approach, for instance. It doesn’t really conflict with the kind of thing Dan was talking about, but it gives a setting for it with lot of abstract machinery. I won’t try to explain the details (which anyway I’m not familiar with), but just enough to suggest how the two seem to me to fit together, after discussing it a bit with Dan. The idea of a cohesive topos seems to start with Bill Lawvere, and it’s supposed to characterize something about those categories which are really “categories of spaces” the way $Top$ is. Intuitively, spaces consist of “points”, which are held together in lumps we could call “pieces”. Hence “cohesion”: the points of a typical space cohere together, rather than being a dust of separate elements. When that happens, in a discrete space, we just say that each piece happens to have just one point in it – but a priori we distinguish the two ideas. So we might normally say that $Top$ has an “underlying set” functor $U : Top \rightarrow Set$, and its left adjoint, the “discrete space” functor $Disc: Set \rightarrow Top$ (left adjoint since set maps from $S$ are the same as continuous maps from $Disc(S)$ – it’s easy for maps out of $Disc(S)$ to be continuous, since every subset is open). In fact, any topos of sheaves on some site has a pair of functors like this (where $U$ becomes $\Gamma$, the “set of global sections” functor), essentially because $Set$ is the topos of sheaves on a single point, and there’s a terminal map from any site into the point. So this adjoint pair is the “terminal geometric morphism” into $Set$. But this omits there are a couple of other things that apply to $Top$: $U$ has a right adjoint, $Codisc: Set \rightarrow Top$, where $Codisc(S)$ has only $S$ and $\emptyset$ as its open sets. In $Codisc(S)$, all the points are “stuck together” in one piece. On the other hand, $Disc$ itself has a left adjoint, $\Pi_0: Top \rightarrow Set$, which gives the set of connected components of a space. $\Pi_0(X)$ is another kind of “underlying set” of a space. So we call a topos $\mathcal{E}$ “cohesive” when the terminal geometric morphism extends to a chain of four adjoint functors in just this way, which satisfy a few properties that characterize what’s happening here. (We can talk about “cohesive sites”, where this happens.) Now $Diff$ isn’t exactly a category of sheaves on a site: it’s the category of concrete sheaves on a (concrete) site. There is a cohesive topos of all sheaves on the diffeological site. (What’s more, it’s known to have a model category structure). But now, it’s a fact that any cohesive topos $\mathcal{E}$ has a subcategory of concrete objects (ones where the canonical unit map $X \rightarrow Codisc(\Gamma(X))$ is mono: roughly, we can characterize the morphisms of $X$ by what they do to its points). This category is always a quasitopos (and it’s a reflective subcategory of $\mathcal{E}$: see the previous post for some comments about reflective subcategories if interested…) This is where $Diff$ fits in here. Diffeologies define a “cohesion” just as topologies do: points are in the same “piece” if there’s some plot from a connected part of $\mathbb{R}^n$ that lands on both. Why is $Diff$ only a quasitopos? Because in general, the subobject classifier in $\mathcal{E}$ isn’t concrete – but it will have a “concretification”, which is the weak subobject classifier I mentioned above. Where the “infinity” part of “infinity-topos” comes in is the connection to homotopy theory. Here, we replace the topos $Sets$ with the infinity-topos of infinity-groupoids. Then the “underlying” functor captures not just the set of points of a space $X$, but its whole fundamental infinity-groupoid. Its objects are points of $X$, its morphisms are paths, 2-morphisms are homotopies of paths, and so on. All the homotopy groups of $X$ live here. So a cohesive inifinity-topos is defined much like above, but with $\infty-Gpd$ playing the role of $Set$, and with that $\Pi_0$ functor replaced by $\Pi$, something which, implicitly, gives all the homotopy groups of $X$. We might look for cohesive infinity-toposes to be given by the (infinity)-categories of simplicial sheaves on cohesive sites. This raises a point Dan made in his talk over the diffeological site $D$, we can talk about a cube of different structures that live over it, starting with presheaves: $PSh(D)$. We can add different modifiers to this: the sheaf condition; the adjective “concrete”; the adjective “simplicial”. Various combinations of these adjectives (e.g. simplicial presheaves) are known to have a model structure. $Diff$ is the case where we have concrete sheaves on $D$. So far, it hasn’t been proved, but it looks like it shortly will be, that this has a model structure. This is a particularly nice one, because these things really do seem a lot like spaces: they’re just sets with some easy-to-define and well-behaved (that’s what the sheaf condition does) structure on them, and they include all the examples a differential geometer requires, the manifolds. One talk at the workshop was nominally a school talk by Laurent Freidel, but it’s interesting and distinctive enough in its own right that I wanted to consider it by itself. It was based on this paper on the “Principle of Relative Locality”. This isn’t so much a new theory, as an exposition of what ought to happen when one looks at a particular limit of any putative theory that has both quantum field theory and gravity as (different) limits of it. This leads through some ideas, such as curved momentum space, which have been kicking around for a while. The end result is a way of accounting for apparently non-local interactions of particles, by saying that while the particles themselves “see” the interactions as local, distant observers might not. Whereas Einstein’s gravity describes a regime where Newton’s gravitational constant $G_N$ is important but Planck’s constant $\hbar$ is negligible, and (special-relativistic) quantum field theory assumes $\hbar$ significant but $G_N$ not. Both of these assume there is a special velocity scale, given by the speed of light $c$, whereas classical mechanics assumes that all three can be neglected (i.e. $G_N$ and $\hbar$ are zero, and $c$ is infinite). The guiding assumption is that these are all approximations to some more fundamental theory, called “quantum gravity” just because it accepts that both $G_N$ and $\hbar$ (as well as $c$) are significant in calculating physical effects. So GR and QFT incorporate two of the three constants each, and classical mechanics incorporates neither. The “principle of relative locality” arises when we consider a slightly different approximation to this underlying theory. This approximation works with a regime where $G_N$ and $\hbar$ are each negligible, but the ratio is not – this being related to the Planck mass $m_p \sim \sqrt{\frac{\hbar}{G_N}}$. The point is that this is an approximation with no special length scale (“Planck length”), but instead a special energy scale (“Planck mass”) which has to be preserved. Since energy and momentum are different parts of a single 4-vector, this is also a momentum scale; we expect to see some kind of deformation of momentum space, at least for momenta that are bigger than this scale. The existence of this scale turns out to mean that momenta don’t add linearly – at least, not unless they’re very small compared to the Planck scale. So what is “Relative Locality”? In the paper linked above, it’s stated like so: Physics takes place in phase space and there is no invariant global projection that gives a description of processes in spacetime. From their measurements local observers can construct descriptions of particles moving and interacting in a spacetime, but different observers construct different spacetimes, which are observer-dependent slices of phase space. Motivation This arises from taking the basic insight of general relativity – the requirement that physical principles should be invariant under coordinate transformations (i.e. diffeomorphisms) – and extend it so that instead of applying just to spacetime, it applies to the whole of phase space. Phase space (which, in this limit where $\hbar = 0$, replaces the Hilbert space of a truly quantum theory) is the space of position-momentum configurations (of things small enough to treat as point-like, in a given fixed approximation). Having no $G_N$ means we don’t need to worry about any dynamical curvature of “spacetime” (which doesn’t exist), and having no Planck length means we can blithely treat phase space as a manifold with coordinates valued in the real line (which has no special scale). Yet, having a special mass/momentum scale says we should see some purely combined “quantum gravity” effects show up. The physical idea is that phase space is an accurate description of what we can see and measure locally. Observers (whom we assume small enough to be considered point-like) can measure their own proper time (they “have a clock”) and can detect momenta (by letting things collide with them and measuring the energy transferred locally and its direction). That is, we “see colors and angles” (i.e. photon energies and differences of direction). Beyond this, one shouldn’t impose any particular theory of what momenta do: we can observe the momenta of separate objects and see what results when they interact and deduce rules from that. As an extension of standard physics, this model is pretty conservative. Now, conventionally, phase space would be the cotangent bundle of spacetime $T^*M$. This model is based on the assumption that objects can be at any point, and wherever they are, their space of possible momenta is a vector space. Being a bundle, with a global projection onto $M$ (taking $(x,v)$ to $x$), is exactly what this principle says doesn’t necessarily obtain. We still assume that phase space will be some symplectic manifold. But we don’t assume a priori that momentum coordinates give a projection whose fibres happen to be vector spaces, as in a cotangent bundle. Now, a symplectic manifold still looks locally like a cotangent bundle (Darboux’s theorem). So even if there is no universal “spacetime”, each observer can still locally construct a version of “spacetime” by slicing up phase space into position and momentum coordinates. One can, by brute force, extend the spacetime coordinates quite far, to distant points in phase space. This is roughly analogous to how, in special relativity, each observer can put their own coordinates on spacetime and arrive at different notions of simultaneity. In general relativity, there are issues with trying to extend this concept globally, but it can be done under some conditions, giving the idea of “space-like slices” of spacetime. In the same way, we can construct “spacetime-like slices” of phase space. Geometrizing Algebra Now, if phase space is a cotangent bundle, momenta can be added (the fibres of the bundle are vector spaces). Some more recent ideas about “quasi-Hamiltonian spaces” (initially introduced by Alekseev, Malkin and Meinrenken) conceive of momenta as “group-valued” – rather than taking values in the dual of some Lie algebra (the way, classically, momenta are dual to velocities, which live in the Lie algebra of infinitesimal translations). For small momenta, these are hard to distinguish, so even group-valued momenta might look linear, but the premise is that we ought to discover this by experiment, not assumption. We certainly can detect “zero momentum” and for physical reasons can say that given two things with two momenta $(p,q)$, there’s a way of combining them into a combined momentum $p \oplus q$. Think of doing this physically – transfer all momentum from one particle to another, as seen by a given observer. Since the same momentum at the observer’s position can be either coming in or going out, this operation has a “negative” with $(\ominus p) \oplus p = 0$. We do have a space of momenta at any given observer’s location – the total of all momenta that can be observed there, and this space now has some algebraic structure. But we have no reason to assume up front that $\oplus$ is either commutative or associative (let alone that it makes momentum space at a given observer’s location into a vector space). One can interpret this algebraic structure as giving some geometry. The commutator for $\oplus$ gives a metric on momentum space. This is a bilinear form which is implicitly defined by the “norm” that assigns a kinetic energy to a particle with a given momentum. The associator given by $p \oplus ( q \oplus r ) - (p \oplus q ) \oplus r)$, infinitesimally near $0$ where this makes sense, gives a connection. This defines a “parallel transport” of a finite momentum $p$ in the direction of a momentum $q$ by saying infinitesimally what happens when adding $dq$ to $p$. Various additional physical assumptions – like the momentum-space “duals” of the equivalence principle (that the combination of momenta works the same way for all kinds of matter regardless of charge), or the strong equivalence principle (that inertial mass and rest mass energy per the relation $E = mc^2$ are the same) and so forth can narrow down the geometry of this metric and connection. Typically we’ll find that it needs to be Lorentzian. With strong enough symmetry assumptions, it must be flat, so that momentum space is a vector space after all – but even with fairly strong assumptions, as with general relativity, there’s still room for this “empty space” to have some intrinsic curvature, in the form of a momentum-space “dual cosmological constant”, which can be positive (so momentum space is closed like a sphere), zero (the vector space case we usually assume) or negative (so momentum space is hyperbolic). This geometrization of what had been algebraic is somewhat analogous to what happened with velocities (i.e. vectors in spacetime)) when the theory of special relativity came along. Insisting that the “invariant” scale $c$ be the same in every reference system meant that the addition of velocities ceased to be linear. At least, it did if you assume that adding velocities has an interpretation along the lines of: “first, from rest, add velocity v to your motion; then, from that reference frame, add velocity w”. While adding spacetime vectors still worked the same way, one had to rephrase this rule if we think of adding velocities as observed within a given reference frame – this became $v \oplus w = (v + w) (1 + uv)$ (scaling so $c =1$ and assuming the velocities are in the same direction). When velocities are small relative to $c$, this looks roughly like linear addition. Geometrizing the algebra of momentum space is thought of a little differently, but similar things can be said: we think operationally in terms of combining momenta by some process. First transfer (group-valued) momentum $p$ to a particle, then momentum $q$ – the connection on momentum space tells us how to translate these momenta into the “reference frame” of a new observer with momentum shifted relative to the starting point. Here again, the special momentum scale $m_p$ (which is also a mass scale since a momentum has a corresponding kinetic energy) is a “deformation” parameter – for momenta that are small compared to this scale, things seem to work linearly as usual. There’s some discussion in the paper which relates this to DSR (either “doubly” or “deformed” special relativity), which is another postulated limit of quantum gravity, a variation of SR with both a special velocity and a special mass/momentum scale, to consider “what SR looks like near the Planck scale”, which treats spacetime as a noncommutative space, and generalizes the Lorentz group to a Hopf algebra which is a deformation of it. In DSR, the noncommutativity of “position space” is directly related to curvature of momentum space. In the “relative locality” view, we accept a classical phase space, but not a classical spacetime within it. Physical Implications We should understand this scale as telling us where “quantum gravity effects” should start to become visible in particle interactions. This is a fairly large scale for subatomic particles. The Planck mass as usually given is about 21 micrograms: small for normal purposes, about the size of a small sand grain, but very large for subatomic particles. Converting to momentum units with $c$, this is about 6 kg m/s: on the order of the momentum of a kicked soccer ball or so. For a subatomic particle this is a lot. This scale does raise a question for many people who first hear this argument, though – that quantum gravity effects should become apparent around the Planck mass/momentum scale, since macro-objects like the aforementioned soccer ball still seem to have linearly-additive momenta. Laurent explained the problem with this intuition. For interactions of big, extended, but composite objects like soccer balls, one has to calculate not just one interaction, but all the various interactions of their parts, so the “effective” mass scale where the deformation would be seen becomes $N m_p$ where $N$ is the number of particles in the soccer ball. Roughly, the point is that a soccer ball is not a large “thing” for these purposes, but a large conglomeration of small “things”, whose interactions are “fundamental”. The “effective” mass scale tells us how we would have to alter the physical constants to be able to treat it as a “thing”. (This is somewhat related to the question of “effective actions” and renormalization, though these are a bit more complicated.) There are a number of possible experiments suggested in the paper, which Laurent mentioned in the talk. One involves a kind of “twin paradox” taking place in momentum space. In “spacetime”, a spaceship travelling a large loop at high velocity will arrive where it started having experienced less time than an observer who remained there (because of the Lorentzian metric) – and a dual phenomenon in momentum space says that particles travelling through loops (also in momentum space) should arrive displaced in space because of the relativity of localization. This could be observed in particle accelerators where particles make several transits of a loop, since the effect is cumulative. Another effect could be seen in astronomical observations: if an observer is observing some distant object via photons of different wavelengths (hence momenta), she might “localize” the object differently – that is, the two photons travel at “the same speed” the whole way, but arrive at different times because the observer will interpret the object as being at two different distances for the two photons. This last one is rather weird, and I had to ask how one would distinguish this effect from a variable speed of light (predicted by certain other ideas about quantum gravity). How to distinguish such effects seems to be not quite worked out yet, but at least this is an indication that there are new, experimentally detectible, effects predicted by this “relative locality” principle. As Laurent emphasized, once we’ve noticed that not accepting this principle means making an a priori assumption about the geometry of momentum space (even if only in some particular approximation, or limit, of a true theory of quantum gravity), we’re pretty much obliged to stop making that assumption and do the experiments. Finding our assumptions were right would simply be revealing which momentum space geometry actually obtains in the approximation we’re studying. A final note about the physical interpretation: this “relative locality” principle can be discovered by looking (in the relevant limit) at a Lagrangian for free particles, with interactions described in terms of momenta. It so happens that one can describe this without referencing a “real” spacetime: the part of the action that allows particles to interact when “close” only needs coordinate functions, which can certainly exist here, but are an observer-dependent construct. The conservation of (non-linear) momenta is specified via a Lagrange multiplier. The whole Lagrangian formalism for the mechanics of colliding particles works without reference to spacetime. Now, even though all the interactions (specified by the conservation of momentum terms) happen “at one location”, in that there will be an observer who sees them happening in the momentum space of her own location. But an observer at a different point may disagree about whether the interaction was local – i.e. happened at a single point in spacetime. Thus “relativity of localization”. Again, this is no more bizarre (mathematically) than the fact that distant, relatively moving, observers in special relativity might disagree about simultaneity, whether two events happened at the same time. They have their own coordinates on spacetime, and transferring between them mixes space coordinates and time coordinates, so they’ll disagree whether the time-coordinate values of two events are the same. Similarly, in this phase-space picture, two different observers each have a coordinate system for splitting phase space into “spacetime” and “energy-momentum” coordinates, but switching between them may mix these two pieces. Thus, the two observers will disagree about whether the spacetime-coordinate values for the different interacting particles are the same. And so, one observer says the interaction is “local in spacetime”, and the other says it’s not. The point is that it’s local for the particles themselves (thinking of them as observers). All that’s going on here is the not-very-astonishing fact that in the conventional picture, we have no problem with interactions being nonlocal in momentum space (particles with very different momenta can interact as long as they collide with each other)… combined with the inability to globally and invariantly distinguish position and momentum coordinates. What this means, philosophically, can be debated, but it does offer some plausibility to the claim that space and time are auxiliary, conceptual additions to what we actually experience, which just account for the relations between bits of matter. These concepts can be dispensed with even where we have a classical-looking phase space rather than Hilbert space (where, presumably, this is even more true). Edit: On a totally unrelated note, I just noticed this post by Alex Hoffnung over at the n-Category Cafe which gives a lot of detail on issues relating to spans in bicategories that I had begun to think more about recently in relation to developing a higher-gauge-theoretic version of the construction I described for ETQFT. In particular, I’d been thinking about how the 2-group analog of restriction and induction for representations realizes the various kinds of duality properties, where we have adjunctions, biadjunctions, and so forth, in which units and counits of the various adjunctions have further duality. This observation seems to be due to Jim Dolan, as far as I can see from a brief note in HDA II. In that case, it’s really talking about the star-structure of the span (tri)category, but looking at the discussion Alex gives suggests to me that this theme shows up throughout this subject. I’ll have to take a closer look at the draft paper he linked to and see if there’s more to say… As usual, this write-up process has been taking a while since life does intrude into blogging for some reason. In this case, because for a little less than a week, my wife and I have been on our honeymoon, which was delayed by our moving to Lisbon. We went to the Azores, or rather to São Miguel, the largest of the nine islands. We had a good time, roughly like so: Now that we’re back, I’ll attempt to wrap up with the summaries of things discussed at the workshop on Higher Gauge Theory, TQFT, and Quantum Gravity. In the previous post I described talks which I roughly gathered under TQFT and Higher Gauge Theory, but the latter really ramifies out in a few different ways. As began to be clear before, higher bundles are classified by higher cohomology of manifolds, and so are gerbes – so in fact these are two slightly different ways of talking about the same thing. I also remarked, in the summary of Konrad Waldorf’s talk, the idea that the theory of gerbes on a manifold is equivalent to ordinary gauge theory on its loop space – which is one way to make explicit the idea that categorification “raises dimension”, in this case from parallel transport of points to that of 1-dimensional loops. Next we’ll expand on that theme, and then finally reach the “Quantum Gravity” part, and draw the connection between this and higher gauge theory toward the end. ## Gerbes and Cohomology The very first workshop speaker, in fact, was Paolo Aschieri, who has done a lot of work relating noncommutative geometry and gravity. In this case, though, he was talking about noncommutative gerbes, and specifically referred to this work with some of the other speakers. To be clear, this isn’t about gerbes with noncommutative group $G$, but about gerbes on noncommutative spaces. To begin with, it’s useful to express gerbes in the usual sense in the right language. In particular, he explain what a gerbe on a manifold $X$ is in concrete terms, giving Hitchin’s definition (viz). A $U(1)$ gerbe can be described as “a cohomology class” but it’s more concrete to present it as: • a collection of line bundles $L_{\alpha \beta}$ associated with double overlaps $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$. Note this gets an algebraic structure (multiplication $\star$ of bundles is pointwise $\otimes$, with an inverse given by the dual, $L^{-1} = L^*$, so we can require… • $L_{\alpha \beta}^{-1} \cong L_{\beta \alpha}$, which helps define… • transition functions $\lambda _{\alpha \beta \gamma}$ on triple overlaps $U_{\alpha \beta \gamma}$, which are sections of $L_{\alpha \beta \gamma} = L_{\alpha \beta} \star L_{\beta \gamma} \star L_{\gamma \alpha}$. If this product is trivial, there’d be a 1-cocycle condition here, but we only insist on the 2-cocycle condition… • $\lambda_{\beta \gamma \delta} \lambda_{\alpha \gamma \delta}^{-1} \lambda_{\alpha \beta \delta} \lambda_{\alpha \beta \gamma}^{-1} = 1$ This is a $U(1)$-gerbe on a commutative space. The point is that one can make a similar definition for a noncommutative space. If the space $X$ is associated with the algebra $A=C^{\infty}(X)$ of smooth functions, then a line bundle is a module for $A$, so if $A$ is noncommutative (thought of as a “space” $X$), a “bundle over $X$ is just defined to be an $A$-module. One also has to define an appropriate “covariant derivative” operator $D$ on this module, and the $\star$-product must be defined as well, and will be noncommutative (we can think of it as a deformation of the $\star$ above). The transition functions are sections: that is, elements of the modules in question. his means we can describe a gerbe in terms of a big stack of modules, with a chosen algebraic structure, together with some elements. The idea then is that gerbes can give an interpretation of cohomology of noncommutative spaces as well as commutative ones. Mauro Spera spoke about a point of view of gerbes based on “transgressions”. The essential point is that an $n$-gerbe on a space $X$ can be seen as the obstruction to patching together a family of $(n-1)$-gerbes. Thus, for instance, a $U(1)$ 0-gerbe is a $U(1)$-bundle, which is to say a complex line bundle. As described above, a 1-gerbe can be understood as describing the obstacle to patching together a bunch of line bundles, and the obstacle is the ability to find a cocycle $\lambda$ satisfying the requisite conditions. This obstacle is measured by the cohomology of the space. Saying we want to patch together $(n-1)$-gerbes on the fibre. He went on to discuss how this manifests in terms of obstructions to string structures on manifolds (already discussed at some length in the post on Hisham Sati’s school talk, so I won’t duplicate here). A talk by Igor Bakovic, “Stacks, Gerbes and Etale Groupoids”, gave a way of looking at gerbes via stacks (see this for instance). The organizing principle is the classification of bundles by the space maps into a classifying space – or, to get the category of principal $G$-bundles on, the category $Top(Sh(X),BG)$, where $Sh(X)$ is the category of sheaves on $X$ and $BG$ is the classifying topos of $G$-sets. (So we have geometric morphisms between the toposes as the objects.) Now, to get further into this, we use that $Sh(X)$ is equivalent to the category of Étale spaces over $X$ – this is a refinement of the equivalence between bundles and presheaves. Taking stalks of a presheaf gives a bundle, and taking sections of a bundle gives a presheaf – and these operations are adjoint. The issue at hand is how to categorify this framework to talk about 2-bundles, and the answer is there’s a 2-adjunction between the 2-category $2-Bun(X)$ of such things, and $Fib(X) = [\mathcal{O}(X)^{op},Cat]$, the 2-category of fibred categories over $X$. (That is, instead of looking at “sheaves of sets”, we look at “sheaves of categories” here.) The adjunction, again, involves talking stalks one way, and taking sections the other way. One hard part of this is getting a nice definition of “stalk” for stacks (i.e. for the “sheaves of categories”), and a good part of the talk focused on explaining how to get a nice tractable definition which is (fibre-wise) equivalent to the more natural one. Bakovic did a bunch of this work with Branislav Jurco, who was also there, and spoke about “Nonabelian Bundle 2-Gerbes“. The paper behind that link has more details, which I’ve yet to entirely absorb, but the essential point appears to be to extend the description of “bundle gerbes” associated to crossed modules up to 2-crossed modules. Bundles, with a structure-group $G$, are classified by the cohomology $H^1(X,G)$ with coefficients in $G$; and whereas “bundle-gerbes” with a structure-crossed-module $H \rightarrow G$ can likewise be described by cohomology $H^1(X,H \rightarrow G)$. Notice this is a bit different from the description in terms of higher cohomology $H^2(X,G)$ for a $G$-gerbe, which can be understood as a bundle-gerbe using the shifted crossed module $G \rightarrow 1$ (when $G$ is abelian. The goal here is to generalize this part to nonabelian groups, and also pass up to “bundle 2-gerbes” based on a 2-crossed module, or crossed complex of length 2, $L \rightarrow H \rightarrow G$ as I described previously for Joao Martins’ talk. This would be classified in terms of cohomology valued in the 2-crossed module. The point is that one can describe such a thing as a bundle over a fibre product, which (I think – I’m not so clear on this part) deals with the same structure of overlaps as the higher cohomology in the other way of describing things. Finally, a talk that’s a little harder to classify than most, but which I’ve put here with things somewhat related to string theory, was Alexander Kahle‘s on “T-Duality and Differential K-Theory”, based on work with Alessandro Valentino. This uses the idea of the differential refinement of cohomology theories – in this case, K-theory, which is a generalized cohomology theory, which is to say that K-theory satisfies the Eilenberg-Steenrod axioms (with the dimension axiom relaxed, hence “generalized”). Cohomology theories, including generalized ones, can have differential refinements, which pass from giving topological to geometrical information about a space. So, while K-theory assigns to a space the Grothendieck ring of the category of vector bundles over it, the differential refinement of K-theory does the same with the category of vector bundles with connection. This captures both local and global structures, which turns out to be necessary to describe fields in string theory – specifically, Ramond-Ramond fields. The point of this talk was to describe what happens to these fields under T-duality. This is a kind of duality in string theory between a theory with large strings and small strings. The talk describes how this works, where we have a manifold with fibres at each point $M\times S^1_r$ with fibres strings of radius $r$ and $M \times S^1_{1/r}$ with radius $1/r$. There’s a correspondence space $M \times S^1_r \times S^1_{1/r}$, which has projection maps down into the two situations. Fields, being forms on such a fibration, can be “transferred” through this correspondence space by a “pull-back and push-forward” (with, in the middle, a wedge with a form that mixes the two directions, $exp( d \theta_r + d \theta_{1/r})$). But to be physically the right kind of field, these “forms” actually need to be representing cohomology classes in the differential refinement of K-theory. ## Quantum Gravity etc. Now, part of the point of this workshop was to try to build, or anyway maintain, some bridges between the kind of work in geometry and topology which I’ve been describing and the world of physics. There are some particular versions of physical theories where these ideas have come up. I’ve already touched on string theory along the way (there weren’t many talks about it from a physicist’s point of view), so this will mostly be about a different sort of approach. Benjamin Bahr gave a talk outlining this approach for our mathematician-heavy audience, with his talk on “Spin Foam Operators” (see also for instance this paper). The point is that one approach to quantum gravity has a theory whose “kinematics” (the description of the state of a system at a given time) is described by “spin networks” (based on $SU(2)$ gauge theory), as described back in the pre-school post. These span a Hilbert space, so the “dynamical” issue of such models is how to get operators between Hilbert spaces from “foams” that interpolate between such networks – that is, what kind of extra data they might need, and how to assign amplitudes to faces and edges etc. to define an operator, which (assuming a “local” theory where distant parts of the foam affect the result independently) will be of the form: $Z(K,\rho,P) = (\prod_f A_f) \prod_v Tr_v(\otimes P_e)$ where $K$ is a particular complex (foam), $\rho$ is a way of assigning irreps to faces of the foam, and $P$ is the assignment of intertwiners to edges. Later on, one can take a discrete version of a path integral by summing over all these $(K, \rho, P)$. Here we have a product over faces and one over vertices, with an amplitude $A_f$ assigned (somehow – this is the issue) to faces. The trace is over all the representation spaces assigned to the edges that are incident to a vertex (this is essentially the only consistent way to assign an amplitude to a vertex). If we also consider spacetimes with boundary, we need some amplitudes $B_e$ at the boundary edges, as well. A big part of the work with such models is finding such amplitudes that meet some nice conditions. Some of these conditions are inherently necessary – to ensure the theory is invariant under gauge transformations, or (formally) changing orientations of faces. Others are considered optional, though to me “functoriality” (that the way of deriving operators respects the gluing-together of foams) seems unavoidable – it imposes that the boundary amplitudes have to be found from the $A_f$ in one specific way. Some other nice conditions might be: that $Z(K, \rho, P)$ depends only on the topology of $K$ (which demands that the $P$ operators be projections); that $Z$ is invariant under subdivision of the foam (which implies the amplitudes have to be $A_f = dim(\rho_f)$). Assuming all these means the only choice is exactly which sub-projection $P_e$ is of the projection onto the gauge-invariant part of the representation space for the faces attached to edge $e$. The rest of the talk discussed this, including some examples (models for BF-theory, the Barrett-Crane model and the more recent EPRL/FK model), and finished up by discussing issues about getting a nice continuum limit by way of “coarse graining”. On a related subject, Bianca Dittrich spoke about “Dynamics and Diffeomorphism Symmetry in Discrete Quantum Gravity”, which explained the nature of some of the hard problems with this sort of discrete model of quantum gravity. She began by asking what sort of models (i.e. which choices of amplitudes) in such discrete models would actually produce a nice continuum theory – since gravity, classically, is described in terms of spacetimes which are continua, and the quantum theory must look like this in some approximation. The point is to think of these as “coarse-graining” of a very fine (perfect, in the limit) approximation to the continuum by a triangulation with a very short length-scale for the edges. Coarse graining means discarding some of the edges to get a coarser approximation (perhaps repeatedly). If the $Z$ happens to be triangulation-independent, then coarse graining makes no difference to the result, nor does the converse process of refining the triangulation. So one question is: if we expect the continuum limit to be diffeomorphism invariant (as is General Relativity), what does this say at the discrete level? The relation between diffeomorphism invariance and triangulation invariance has been described by Hendryk Pfeiffer, and in the reverse direction by Dittrich et al. Actually constructing the dynamics for a system like this in a nice way (“canonical dynamics with anomaly-free constraints”) is still a big problem, which Bianca suggested might be approached by this coarse-graining idea. Now, if a theory is topological (here we get the link to TQFT), such as electromagnetism in 2D, or (linearized) gravity in 3D, coarse graining doesn’t change much. But otherwise, changing the length scale means changing the action for the continuum limit of the theory. This is related to renormalization: one starts with a “naive” guess at a theory, then refines it (in this case, by the coarse-graining process), which changes the action for the theory, until arriving at (or approximating to) a fixed point. Bianca showed an example, which produces a really huge, horrible action full of very complicated terms, which seems rather dissatisfying. What’s more, she pointed out that, unless the theory is topological, this always produces an action which is non-local – unlike the “naive” discrete theory. That is, the action can’t be described in terms of a bunch of non-interacting contributions from the field at individual points – instead, it’s some function which couples the field values at distant points (albeit in a way that falls off exponentially as the points get further apart). In a more specific talk, Aleksandr Mikovic discussed “Finiteness and Semiclassical Limit of EPRL-FK Spin Foam Models”, looking at a particular example of such models which is the (relatively) new-and-improved candidate for quantum gravity mentioned above. This was a somewhat technical talk, which I didn’t entirely follow, but roughly, the way he went at this was through the techniques of perturbative QFT. That is, by looking at the theory in terms of an “effective action”, instead of some path integral over histories $\phi$ with action $S(\phi)$ – which looks like $\int d\phi e^{iS(\phi)}$. Starting with some classical history $\bar{\phi}$ – a stationary point of the action $S$ – the effective action $\Gamma(\bar{\phi})$ is an integral over small fluctuations $\phi$ around it of $e^{iS(\bar{\phi} + \phi)}$. He commented more on the distinction between the question of triangulation independence (which is crucial for using spin foams to give invariants of manifolds) and the question of whether the theory gives a good quantum theory of gravity – that’s the “semiclassical limit” part. (In light of the above, this seems to amount to asking if “diffeomorphism invariance” really extends through to the full theory, or is only approximately true, in the limiting case). Then the “finiteness” part has to do with the question of getting decent asymptotic behaviour for some of those weights mentioned above so as to give a nice effective action (if not necessarily triangulation independence). So, for instance, in the Ponzano-Regge model (which gives a nice invariant for manifolds), the vertex amplitudes $A_v$ are found by the 6j-symbols of representations. The asymptotics of the 6j symbols then becomes an issue – Alekandr noted that to get a theory with a nice effective action, those 6j-symbols need to be scaled by a certain factor. This breaks triangulation independence (hence means we don’t have a good manifold invariant), but gives a physically nicer theory. In the case of 3D gravity, this is not what we want, but as he said, there isn’t a good a-priori reason to think it can’t give a good theory of 4D gravity. Now, making a connection between these sorts of models and higher gauge theory, Aristide Baratin spoke about “2-Group Representations for State Sum Models”. This is a project Baez, Freidel, and Wise, building on work by Crane and Sheppard (see my previous post, where Derek described the geometry of the representation theory for some 2-groups). The idea is to construct state-sum models where, at the kinematical level, edges are labelled by 2-group representations, faces by intertwiners, and tetrahedra by 2-intertwiners. (This assumes the foam is a triangulation – there’s a certain amount of back-and-forth in this area between this, and the Poincaré dual picture where we have 4-valent vertices). He discussed this in a couple of related cases – the Euclidean and Poincaré 2-groups, which are described by crossed modules with base groups $SO(4)$ or $SO(3,1)$ respectively, acting on the abelian group (of automorphisms of the identity) $R^4$ in the obvious way. Then the analogy of the 6j symbols above, which are assigned to tetrahedra (or dually, vertices in a foam interpolating two kinematical states), are now 10j symbols assigned to 4-simplexes (or dually, vertices in the foam). One nice thing about this setup is that there’s a good geometric interpretation of the kinematics – irreducible representations of these 2-groups pick out orbits of the action of the relevant $SO$ on $R^4$. These are “mass shells” – radii of spheres in the Euclidean case, or proper length/time values that pick out hyperboloids in the Lorentzian case of $SO(3,1)$. Assigning these to edges has an obvious geometric meaning (as a proper length of the edge), which thus has a continuous spectrum. The areas and volumes interpreting the intertwiners and 2-intertwiners start to exhibit more of the discreteness you see in the usual formulation with representations of the $SO$ groups themselves. Finally, Aristide pointed out that this model originally arose not from an attempt to make a quantum gravity model, but from looking at Feynman diagrams in flat space (a sort of “quantum flat space” model), which is suggestively interesting, if not really conclusively proving anything. Finally, Laurent Freidel gave a talk, “Classical Geometry of Spin Network States” which was a way of challenging the idea that these states are exclusively about “quantum geometries”, and tried to give an account of how to interpret them as discrete, but classical. That is, the quantization of the classical phase space $T^*(A/G)$ (the cotangent bundle of connections-mod-gauge) involves first a discretization to a spin-network phase space $\mathcal{P}_{\Gamma}$, and then a quantization to get a Hilbert space $H_{\Gamma}$, and the hard part is the first step. The point is to see what the classical phase space is, and he describes it as a (symplectic) quotient $T^*(SU(2)^E)//SU(2)^V$, which starts by assigning$T^*(SU(2))$to each edge, then reduced by gauge transformations. The puzzle is to interpret the states as geometries with some discrete aspect. The answer is that one thinks of edges as describing (dual) faces, and vertices as describing some polytopes. For each $p$, there’s a $2(p-3)$-dimensional “shape space” of convex polytopes with $p$-faces and a given fixed area $j$. This has a canonical symplectic structure, where lengths and interior angles at an edge are the canonically conjugate variables. Then the whole phase space describes ways of building geometries by gluing these things (associated to vertices) together at the corresponding faces whenever the two vertices are joined by an edge. Notice this is a bit strange, since there’s no particular reason the faces being glued will have the same shape: just the same area. An area-1 pentagon and an area-1 square associated to the same edge could be glued just fine. Then the classical geometry for one of these configurations is build of a bunch of flat polyhedra (i.e. with a flat metric and connection on them). Measuring distance across a face in this geometry is a little strange. Given two points inside adjacent cells, you measure orthogonal distance to the matched faces, and add in the distance between the points you arrive at (orthogonally) – assuming you glued the faces at the centre. This is a rather ugly-seeming geometry, but it’s symplectically isomorphic to the phase space of spin network states – so it’s these classical geometries that spin-foam QG is a quantization of. Maybe the ugliness should count against this model of quantum gravity – or maybe my aesthetic sense just needs work. (Laurent also gave another talk, which was originally scheduled as one of the school talks, but ended up being a very interesting exposition of the principle of “Relativity of Localization”, which is hard to shoehorn into the themes I’ve used here, and was anyway interesting enough that I’ll devote a separate post to it.) I’d like to continue describing the talks that made up the HGTQGR workshop, in particular the ones that took place during the school portion of the event. I’ll save one “school” session, by Laurent Freidel, to discuss with the talks because it seems to more nearly belong there. This leaves five people who gave between two and four lectures each over a period of a few days, all intermingled. Here’s a very rough summary in the order of first appearance: ## 2D Extended TQFT Chris Schommer-Pries gave the longest series of talks, about the classification of 2D extended TQFT’s. A TQFT is a kind of topological invariant for manifolds, which has a sort of “locality” property, in that you can decompose the manifold, compute the invariant on the parts, and find the whole by gluing the pieces back together. This is expressed by saying it’s a monoidal functor $Z : (Cob_d, \sqcup) \rightarrow (Vect, \otimes)$, where the “locality” property is now functoriality property that composition is preserved. The key thing here is the cobordism category $Cob_d$, which has objects (d-1)-dimensional manifolds, and morphisms d-dimensional cobordisms (manifolds with boundary, where the objects are components of the boundary). Then a closed d-manifold is just a cobordism from$latex\emptyset\$ to itself.

Making this into a category is actually a bit nontrivial: gluing bits of smooth manifolds, for instance, won’t necessarily give something smooth.  There are various ways of handling this, such as giving the boundaries “collars”, but Chris’ preferred method is to give boundaries (and, ultimately, corners, etc.) a”halation”.  This word originally means the halo of light around bright things you sometimes see in photos, but in this context, a halation for $X$ is an equivalence class of embeddings into neighborhoods $U \subset \mathbb{R}^d$.  The equivalence class says two such embeddings into $U$ and $V$ are equivalent if there’s a compatible refinement into some common $W$ that embeds into both $U$ and $V$.  The idea is that a halation is a kind of d-dimensional “halo”, or the “germ of a d-manifold” around $X$.  Then gluing compatibly along (d-1)-boundaries with halations ensures that we get smooth d-manifolds.  (One can also extend this setup so that everything in sight is oriented, or has some other such structure on it.)

In any case, an extended TQFT will then mean an n-functor $Z : (Bord_d,\sqcup) \rightarrow (\mathcal{C},\otimes)$, where $(\mathcal{C},\otimes)$ is some symmetric monoidal n-category (which is supposed to be similar to $Vect$).  Its exact nature is less important than that of $Bord_d$, which has:

• 0-Morphisms (i.e. Objects): 0-manifolds (collections of points)
• 1-Morphisms: 1-dimensional cobordisms between 0-manifolds (curves)
• 2-Morphisms: 2-dim cobordisms with corners between 1-Morphisms (surfaces with boundary)
• d-Morphisms: d-dimensional cobordisms between (d-1)-Morphisms (n-manifolds with corners), up to isomorphism

(Note: the distinction between “Bord” and “Cobord” is basically a matter of when a given terminology came in.  “Cobordism” and “Bordism”, unfortunately, mean the same thing, except that “bordism” has become popular more recently, since the “co” makes it sound like it’s the opposite category of something else.  This is kind of regrettable, but that’s what happened.  Sorry.)

The crucial point, is that Chris wanted to classify all such things, and his approach to this is to give a presentation of $Bord_d$.  This is based on stuff in his thesis.  The basic idea is to use Morse theory, and its higher-dimensional generalization, Cerf theory.  The idea is that one can put a Morse function  on a cobordism (essentially, a well-behaved “time order” on points) and look at its critical points.  Classifying these tells us what the generators for the category of cobordisms must be: there need to be enough to capture all the most general sorts of critical points.

Cerf theory does something similar, but one dimension up: now we’re talking about “stratified” families of Morse functions.  Again one studies critical points, but, for instance, on a 2-dim surface, there can be 1- and 0-dimensional parts of the set of cricical points.  In general, this gets into the theory of higher-dimensional singularities, catastrophe theory, and so on.  Each extra dimension one adds means looking at how the sets of critical points in the previous dimension can change over “time” (i.e. within some stratified family of Cerf functions).  Where these changes themselves go through critical points, one needs new generators for the various j-morphisms of the cobordism category.  (See some examples of such “catastrophes”, such as folds, cusps, swallowtails, etc. linked from here, say.)  Showing what such singularities can be like in the “generic” situation, and indeed, even defining “generic” in a way that makes sense in any dimension, required some discussion of jet bundles.  These are generalizations of tangent bundles that capture higher derivatives the way tangent bundles capture first-derivatives.  The essential point is that one can find a way to decompose these into a direct sum of parts of various dimensions (capturing where various higher derivatives are zero, say), and these will eventually tell us the dimension of a set of critical points for a Cerf function.

Now, this gives a characterization of what cobordisms can be like – part of the work in the theorem is to show that this is sufficient: that is, given a diagram showing the critical points for some Morse/Cerf function, one needs to be able to find the appropriate generators and piece together the cobordism (possibly a closed manifold) that it came from.  Chris showed how this works – a slightly finicky process involving cutting a diagram of the singular points (with some extra labelling information) into parts, and using a graphical calculus to work out how pasting works – and showed an example reconstruction of a surface this way.  This amounts to a construction of an equivalence between an “abstract” cobordism category given in terms of generators (and relations) which come from Cerf theory, and the concrete one.  The theorem then says that there’s a correspondence between equivalence classes of 2D cobordisms, and certain planar diagrams, up to some local moves.  To show this properly required a digression through some theory of symmetric monoidal bicategories, and what the right notion of equivalence for them is.

This all done, the point is that $Bord_d$ has a characterization in terms of a universal property, and so any ETQFT $Z : Bord_d \rightarrow \mathcal{C}$ amounts to a certain kind of object in $\mathcal{C}$ (corresponding to the image of the point – the generating object in $Bord_d$).  For instance, in the oriented situation this object needs to be “fully dualizable”: it should have a dual (the point with opposite orientation), and a whole bunch of maps that specify the duality: a cobordism from $(+,-)$ to nothing (just the “U”-shaped curve), which has a dual – and some 2-D cobordisms which specify that duality, and so on.  Specifying all this dualizability structure amounts to giving the image of all the generators of cobordisms, and determines the functors $Z$, and vice versa.

This is a rapid summary of six hours of lectures, of course, so for more precise versions of these statements, you may want to look into Chris’ thesis as linked above.

## Homotopy QFT and the Crossed Menagerie

The next series of lectures in the school was Tim Porter’s, about relations between Homotopy Quantum Field Theory (HQFT) and various sort of crossed gizmos.  HQFT is an idea introduced by Vladimir Turaev, (see his paper with Tim here, for an intro, though Turaev also now has a book on the subject).  It’s intended to deal with similar sorts of structures to TQFT, but with various sorts of extra structure.  This structure is related to the “Crossed Menagerie”, on which Tim has written an almost unbelievably extensive bunch of lecture notes, of which a special short version was made for this lecture series that’s a mere 350 pages long.

Anyway, the cobordism category $Bord_d$ described above is replaced by one Tim called $HCobord(d,B)$ (see above comment about “bord” and “cobord”, which mean the same thing).  Again, this has d-dimensional cobordisms as its morphisms and (d-1)-dimensional manifolds as its objects, but now everything in sight is equipped with a map into a space $B$ – almost.  So an object is $X \rightarrow B$, and a morphism is a cobordism with a homotopy class of maps $M \rightarrow B$ which are compatible with the ones at the boundaries.  Then just as a d-TQFT is a representation (i.e. a functor) of $Cob_d$ into $Vect$, a $(d,B)$-HQFT is a representation of $HCobord(d,B)$.

The motivating example here is when $B = B(G)$, the classifying space of a group.  These spaces are fairly complicated when you describe them as built from gluing cells (in homotopy theory, one typically things of spaces as something like CW-complexes: a bunch of cells in various dimensions glued together with face maps etc.), but $B(G)$ has the property that its fundamental group is $G$, and all other homotopy groups are trivial (ensuring this part is what makes the cellular decomposition description tricky).

The upshot is that there’s a correspondence between (homotopy classes of) maps $Map(X ,B(G)) \simeq Hom(\pi(X),G)$ (this makes a good alternative definition of the classifying space, though one needs to ).  Since a map from the fundamental group into $G$ amounts to a flat principal $G$-bundle, we can say that $HCobord(d,B(G))$ is a category of manifolds and cobordisms carrying such a bundle.  This gets us into gauge theory.

But we can go beyond and into higher gauge theory (and other sorts of structures) by picking other sorts of $B$.  To begin with, notice that the correspondence above implies that mapping into $B(G)$ means that when we take maps up to homotopy, we can only detect the fundamental group of $X$, and not any higher homotopy groups.  We say we can only detect the “homotopy 1-type” of the space.  The “homotopy n-type” of a given space $X$ is just the first $n$ homotopy groups $(\pi_1(X), \dots, \pi_n(X))$.  Alternatively, an “n-type” is an equivalence class of spaces which all have the same such groups.  Or, again, an “n-type” is a particular representative of one of these classes where these are the only nonzero homotopy groups.

The point being that if we’re considering maps $X \rightarrow B$ up to homotopy, we may only be detecting the n-type of $X$ (and therefore may as well assume $X$ is an n-type in the last sense when it’s convenient).  More precisely, there are “Postnikov functors” $P_n(-)$ which take a space $X$ and return the corresponding n-type.  This can be done by gluing in “patches” of higher dimensions to “fill in the holes” which are measured by the higher homotopy groups (in general, the result is infinite dimensional as a cell complex).  Thus, there are embeddings $X \hookrightarrow P_n(X)$, which get along with the obvious chain

$\dots \rightarrow P_{n+1}(X) \rightarrow P_n(X) \rightarrow P_{n-1}(X) \rightarrow \dots$

There was a fairly nifty digression here explaining how this is a “coskeleton” of $X$, in that $P_n$ is a right adjoint to the “n-skeleton” functor (which throws away cells above dimension n, not homotopy groups), so that $S(Sk_n(M),X) \cong S(M,P_n(X))$.  To really explain it properly, though I would have to really explain what that $S$ is (it refers to maps in the category of simplicial sets, which are another nice model of spaces up to homotopy).  This digression would carry us away from higher gauge theory, which is where I’m going.

One thing to say is that if $X$ is d-dimensional, then any HQFT is determined entirely by the d-type of $B$.  Any extra jazz going on in $B$‘s higher homotopy groups won’t be detected when we’re only mapping a d-dimensional space $X$ into it.  So one might as well assume that $B$ is just a d-type.

We want to say we can detect a homotopy n-type of a space if, for example, $B = B(\mathcal{G})$ where $\mathcal{G}$ is an “n-group”.  A handy way to account for this is in terms of a “crossed complex”.  The first nontrivial example of this would be a crossed module, which consists of

• Two groups, $G$ and $H$ with
• A map $\partial : H \rightarrow G$ and
• An action of $G$ on $H$ by automorphisms, $G \rhd H$
• all such that action looks as much like conjugation as possible:
• $\partial(g \rhd h) = g (\partial h) g^{-1}$ (so that $\partial$ is $G$-equivariant)
• $\partial h \rhd h' = h h' h^{-1}$ (the “Peiffer identity”)

This definition looks a little funny, but it does characterize “2-groups” in the sense of categories internal to $\mathbf{Groups}$ (the definition used elsewhere), by taking $G$ to be the group of objects, and $H$ the group of automorphisms of the identity of $G$.  In the description of John Huerta’s lectures, I’ll get back to how that works.

The immediate point is that there are a bunch of natural examples of crossed modules.  For instance: from normal subgroups, where $\partial: H \subset G$ is inclusion and the action really is conjugation; from fibrations, using fundamental groups of base and fibre; from a canonical case where $H = Aut(G)$  and $\partial = 1$ takes everything to the identity; from modules, taking $H$ to be a $G$-module as an abelian group and $\partial = 1$ again.  The first and last give the classical intuition of these guys: crossed modules are simultaneous generalizations of (a) normal subgroups of $G$, and (b) $G$-modules.

There are various other examples, but the relevant thing here is a theorem of MacLane and Whitehead, that crossed modules model all connected homotopy 2-types.  That is, there’s a correspondence between crossed modules up to isomorphism and 2-types.  Of course, groups model 1-types: any group is the fundmental group for a 1-type, and any 1-type is the classifying space for some group.  Likewise, any crossed module determines a 2-type, and vice versa.  So this theorem suggests why crossed modules might deserve to be called “2-groups” even if they didn’t naturally line up with the alternative definition.

To go up to 3-types and 4-types, the intuitive idea is: “do for crossed modules what we did for groups”.  That is, instead of a map of groups $\partial : H \rightarrow G$, we consider a map of crossed modules (which is given by a pair of maps between the groups in each) and so forth.  The resulting structure is a square diagram in $\mathbf{Groups}$ with a bunch of actions.  Each of these maps is the $\partial$ map for a crossed module.  (We can think of the normal subgroup situation: there are two normal subgroups $H,K$ of $G$, and in each of them, the intersection $H \cap K$ is normal, so it determines a crossed module).  This is a “crossed square”, and things like this correspond exactly to homotopy 3-types.  This works roughly as before, since there is a notion of a classifying space $B(\mathcal{G})$ where $\mathcal{G} = (G,H,\partial,\rhd)$, and similarly on for crossed n-cubes.   We can carry on in this way to define a “crossed n-cube”, which correspond to homotopy (n+1)-types.  The correspondence is a little bit more fiddly than it was for groups, but it still exists: any (n+1)-type is the classifying space for a crossed n-cube, and any such crossed n-cube has an (n+1)-type for its classifying space.

This correspondence is the point here.  As we said, when looking at HQFT’s from $HCobord(d,B)$, we may as well assume that $B$ is a d-type.  But then, it’s a classifying space for some crossed (d-1)-cube.  This is a sensible sort of $B$ to use in an HQFT, and it ends up giving us a theory which is related to higher gauge theory: a map $X \rightarrow B(\mathcal{G})$ up to homotopy, where $\mathcal{G}$ is a crossed n-cube will correspond to the structure of a flat $(n+1)$-bundle on $X$, and similarly for cobordisms.  HQFT’s let us look at the structure of this structured cobordism category by means of its linear representations.  Now, it may be that this crossed-cube point of view isn’t the best way to look at $B$, but it is there, and available.

To say more about this, I’ll have to talk more directly about higher gauge theory in its own terms – which I’ll do in part IIb, since this is already pretty long.

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