### homotopy theory

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.

Well, it’s been a while, but it’s now a new semester here in Hamburg, and I wanted to go back and look at some of what we talked about in last semester’s research seminar. This semester, Susama Agarwala and I are sharing the teaching in a topics class on “Category Theory for Geometry“, in which I’ll be talking about categories of sheaves, and building up the technology for Susama to talk about Voevodsky’s theory of motives (enough to give a starting point to read something like this).

As for last semester’s seminar, one of the two main threads, the one which Alessandro Valentino and I helped to organize, was a look at some of the material needed to approach Jacob Lurie’s paper on the classification of topological quantum field theories. The idea was for the research seminar to present the basic tools that are used in that paper to a larger audience, mostly of graduate students – enough to give a fairly precise statement, and develop the tools needed to follow the proof. (By the way, for a nice and lengthier discussion by Chris Schommer-Pries about this subject, which includes more details on much of what’s in this post, check out this video.)

So: the key result is a slightly generalized form of the Cobordism Hypothesis.

### Cobordism Hypothesis

The sort of theory which the paper classifies are those which “extend down to a point”. So what does this mean? A topological field theory can be seen as a sort of “quantum field theory up to homotopy”, which abstract away any geometric information about the underlying space where the fields live – their local degrees of freedom.  We do this by looking only at the classes of fields up to the diffeomorphism symmetries of the space.  The local, geometric, information gets thrown away by taking this quotient of the space of solutions.

In spite of reducing the space of fields this way, we want to capture the intuition that the theory is still somehow “local”, in that we can cut up spaces into parts and make sense of the theory on those parts separately, and determine what it does on a larger space by gluing pieces together, rather than somehow having to take account of the entire space at once, indissolubly. This reasoning should apply to the highest-dimensional space, but also to boundaries, and to any figures we draw on boundaries when cutting them up in turn.

Carrying this on to the logical end point, this means that a topological quantum field theory in the fully extended sense should assign some sort of data to every geometric entity from a zero-dimensional point up to an $n$-dimensional cobordism.  This is all expressed by saying it’s an $n$-functor:

$Z : Bord^{fr}_n(n) \rightarrow nAlg$.

Well, once we know what this means, we’ll know (in principle) what a TQFT is.  It’s less important, for the purposes of Lurie’s paper, what $nAlg$ is than what $Bord^){fr}_n(n)$ is.  The reason is that we want to classify these field theories (i.e. functors).  It will turn out that $Bord_n(n)$ has the sort of structure that makes it easy to classify the functors out of it into any target $n$-category $\mathcal{C}$.  A guess about what kind of structure is actually there was expressed by Baez and Dolan as the Cobordism Hypothesis.  It’s been slightly rephrased from the original form to get a form which has a proof.  The version Lurie proves says:

The $(\infty,n)$-category $Bord^{fr}_n(n)$ is equivalent to the free symmetric monoidal $(\infty,n)$-category generated by one fully-dualizable object.

The basic point is that, since $Bord^{fr}_n(n)$ is a free structure, the classification means that the extended TQFT’s amount precisely to the choice of a fully-dualizable object of $\mathcal{C}$ (which includes a choice of a bunch of morphisms exhibiting the “dualizability”). However, to make sense of this, we need to have a suitable idea of an $(\infty,n)$-category, and know what a fully dualizable object is. Let’s begin with the first.

### $(\infty,n)$-Categories

In one sense, the Cobordism Hypothesis, which was originally made about $n$-categories at a time when these were only beginning to be defined, could be taken as a criterion for an acceptable definition. That is, it expressed an intuition which was important enough that any definition which wouldn’t allow one to prove the Cobordism Hypothesis in some form ought to be rejected. To really make it work, one had to bring in the “infinity” part of $(\infty,n)$-categories. The point here is that we are talking about category-like structures which have morphisms between objects, 2-morphisms between morphisms, and so on, with $j$-morphisms between $j-1$-morphisms for every possible degree. The inspiration for this comes from homotopy theory, where one has maps, homotopies of maps, homotopies of homotopies, etc.

Nowadays, there are several possible concrete models for $(\infty,n)$-categories (see this survey article by Julie Bergner for a summary of four of them). They are all equivalent definitions, in a suitable up-to-homotopy way, but for purposes of the proof, Lurie is taking the definition that an $(\infty,n)$-category is an n-fold complete Segal space. One theme that shows up in all the definitions is that of simplicial methods. (In our seminar, we started with a series of two talks introducing the notions of simplicial sets, simplicial objects in a category, and Kan complexes. If you don’t already know this, essentially everything we need is nicely explained in here.)

One of the underlying ideas is that a category $C$ can be associated with a simplicial set, its nerve $N(C)_{\bullet}$, where the set $N(C)_k$ of $k$-dimensional simplexes is just the set of composable $k$-tuples of morphisms in $C$. If $C$ is a groupoid (everything is invertible), then the simplicial set is a Kan complex – it satisfies some filling conditions, which ensure that any morphism has an inverse. Not every Kan complex is the nerve of a groupoid, but one can think of them as weak versions of groupoids – $\infty$-groupoids, or $(\infty,0)$-categories – where the higher morphisms may not be completely trivial (as with a groupoid), but where at least they’re all invertible. This leads to another desirable feature in any definition of $(\infty,n)$-category, which is the Homotopy Hypothesis: that the $(\infty,1)$-category of $(\infty,0)$-categories, also called $\infty$-groupoids, should be equivalent (in the same weak sense) to a category of Hausdorff spaces with some other nice properties, which we call $\mathbf{Top}$ for short. This is true of Kan complexes.

Thus, up to homotopy, specifying an $\infty$-groupoid is the same as specifying a space.

The data which defines a Segal space (which was however first explicitly defined by Charlez Rezk) is a simplicial space $X_{\bullet}$: for each $n$, there are spaces $X_n$, thought of as the space of composable $n$-tuples of morphisms. To keep things tame, we suppose that $X_0$, the space of objects, is discrete – that is, we have only a set of objects. Being a simplicial space means that the $X_n$ come equipped with a collection of face maps $d_i : X_n \rightarrow X_{n-1}$, which we should think of as compositions: to get from an $n$-tuple to an $(n-1)$-tuple of morphisms, one can compose two morphisms together at any of $(n-1)$ positions in the tuple.

One condition which a simplicial space has to satisfy to be a Segal space has to do with the “weakening” which makes a Segal space a weaker notion than just a category lies in the fact that the $X_n$ cannot be arbitrary, but must be homotopy equivalent to the “actual” space of $n$-tuples, which is a strict pullback $X_1 \times_{X_0} \dots \times_{X_0} X_1$. That is, in a Segal space, the pullback which defines these tuples for a category is weakened to be a homotopy pullback. Combining this with the various face maps, we therefore get a weakened notion of composition: $X_1 \times_{X_0} \dots \times_{X_0} X_1 \cong X_n \rightarrow X_1$. Because we start by replacing the space of $n$-tuples with the homotopy-equivalent $X_n$, the composition rule will only satisfy all the relations which define composition (associativity, for instance) up to homotopy.

To be complete, the Segal space must have a notion of equivalence for $X_{\bullet}$ which agrees with that for Kan complexes seen as $\infty$-groupoids. In particular, there is a sub-simplicial object $Core(X_{\bullet})$, which we understand to consist of the spaces of invertible $k$-morphisms. Since there should be nothing interesting happening above the top dimension, we ask that, for these spaces, the face and degeneracy maps are all homotopy equivalences: up to homotopy, the space of invertible higher morphisms has no new information.

Then, an $n$-fold complete Segal space is defined recursively, just as one might define $n$-categories (without the infinitely many layers of invertible morphisms “at the top”). In that case, we might say that a double category is just a category internal to $\mathbf{Cat}$: it has a category of objects, and a category of morphims, and the various maps and operations, such as composition, which make up the definition of a category are all defined as functors. That turns out to be the same as a structure with objects, horizontal and vertical morphisms, and square-shaped 2-cells. If we insist that the category of objects is discrete (i.e. really just a set, with no interesting morphisms), then the result amounts to a 2-category. Then we can define a 3-category to be a category internal to $\mathbf{2Cat}$ (whose 2-category of objects is discrete), and so on. This approach really defines an $n$-fold category (see e.g. Chapter 5 of Cheng and Lauda to see a variation of this approach, due to Tamsamani and Simpson), but imposing the condition that the objects really amount to a set at each step gives exactly the usual intuition of a (strict!) $n$-category.

This is exactly the approach we take with $n$-fold complete Segal spaces, except that some degree of weakness is automatic. Since a C.S.S. is a simplicial object with some properties (we separately define objects of $k$-tuples of morphisms for every $k$, and all the various composition operations), the same recursive approach leads to a definition of an “$n$-fold complete Segal space” as simply a simplicial object in $(n-1)$-fold C.S.S.’s (with the same properties), such that the objects form a set. In principle, this gives a big class of “spaces of morphisms” one needs to define – one for every $n$-fold product of simplexes of any dimension – but all those requirements that any space of objects “is just a set” (i.e. is homotopy-equivalent to a discrete set of points) simplifies things a bit.

### Cobordism Category as $(\infty,n)$-Category

So how should we think of cobordisms as forming an $(\infty,n)$-category? There are a few stages in making a precise definition, but the basic idea is simple enough. One starts with manifolds and cobordisms embedded in some fixed finite-dimensional vector space $V \times \mathbb{R}^n$, and then takes a limit over all $V$. In each $V \times \mathbb{R}^n$, the coordinates of the $\mathbb{R}^n$ factor give $n$ ways of cutting the cobordism into pieces, and gluing them back together defines composition in a different direction. Now, this won’t actually produce a complete Segal space: one has to take a certain kind of completion. But the idea is intuitive enough.

We want to define an $n$-fold C.S.S. of cobordisms (and cobordisms between cobordisms, and so on, up to $n$-morphisms). To start with, think of the case $n=1$: then the space of objects of $Bord^{fr}_1(1)$ consists of all embeddings of a $(d-1)$-dimensional manifold into $V$. The space of $k$-simplexes (of $k$-tuples of morphisms) consists of all ways of cutting up a $d$-dimensional cobordism embedded in $V \times \mathbb{R}$ by choosing $t_0, \dots , t_{k-2}$, where we think of the cobordism having been glued from two pieces, where at the slice $V \times {t_i}$, we have the object where the two pieces were composed. (One has to be careful to specify that the Morse function on the cobordisms, got by projection only $\mathbb{R}$, has its critical points away from the $t_i$ – the generic case – to make sure that the objects where gluing happens are actual manifolds.)

Now, what about the higher morphisms of the $(\infty,1)$-category? The point is that one needs to have an $\infty$-groupoid – that is, a space! – of morphisms between two cobordisms $M$ and $N$. To make sense of this, we just take the space $Diff(M,N)$ of diffeomorphisms – not just as a set of morphisms, but including its topology as well. The higher morphisms, therefore, can be thought of precisely as paths, homotopies, homotopies between homotopies, and so on, in these spaces. So the essential difference between the 1-category of cobordisms and the $(\infty,1)$-category is that in the first case, morphisms are diffeomorphism classes of cobordisms, whereas in the latter, the higher morphisms are made precisely of the space of diffeomorphisms which we quotient out by in the first case.

Now, $(\infty,n)$-categories, can have non-invertible morphisms between morphisms all the way up to dimension $n$, after which everything is invertible. An $n$-fold C.S.S. does this by taking the definition of a complete Segal space and copying it inside $(n-1)$-fold C.S.S’s: that is, one has an $(n-1)$-fold Complete Segal Space of $k$-tuples of morphisms, for each $k$, they form a simplicial object, and so forth.

Now, if we want to build an $(\infty,n)$-category $Bord^{fr}_n(n)$ of cobordisms, the idea is the same, except that we have a simplicial object, in a category of simplicial objects, and so on. However, the way to define this is essentially similar. To specify an $n$-fold C.S.S., we have to specify a whole collection of spaces associated to cobordisms equipped with embeddings into $V \times \mathbb{R}^n$. In particular, for each tuple $(k_1,\dots,k_n)$, we have the space of such embeddings, such that for each $i = 1 \dots n$ one has $k_i$ special points $t_{i,j}$ along the $i^{th}$ coordinate axis. These are the ways of breaking down a given cobordism into a composite of $k_i +1$ pieces. Again, one has to make sure that these critical points of the Morse functions defined by the projections onto these coordinate axes avoid these special $t_{i,j}$ which define the manifolds where gluing takes place. The composition maps which make these into a simplical object are quite natural – they just come by deleting special points.

Finally, we take a limit over all $V$ (to get around limits to embeddings due to the dimension of $V$). So we know (at least abstractly) what the $(\infty,n)$-category of cobordisms should be. The cobordism hypothesis claims it is equivalent to one defined in a free, algebraically-flavoured way, namely as the free symmetric monoidal $(\infty,n)$-category on a fully-dualizable object. (That object is “the point” – which, up to the kind of homotopically-flavoured equivalence that matters here, is the only object when our highest-dimensional cobordisms have dimension $n$).

### Dualizability

So what does that mean, a “fully dualizable object”?

First, to get the idea, let’s think of the 1-dimensional example.  Instead of “$(\infty,n)$-category”, we would like to just think of this as a statement about a category.  Then $Bord^{fr}_1(1)$ is the 1-category of framed bordisms. For a manifold (or cobordism, which is a manifold with boundary), a framing is a trivialization of the tangent bundle.  That is, it amounts to a choice of isomorphism at each point between the tangent space there and the corresponding $\mathbb{R}^n$.  So the objects of $Bord^{fr}_1(1)$ are collections of (signed) points, and the morphisms are equivalence classes of framed 1-dimensional cobordisms.  These amount to oriented 1-manifolds with boundary, where the points (objects) on the boundary are the source and target of the cobordism.

Now we want to classify what TQFT’s live on this category.  These are functors $Z : Bord^{fr}_1(1)$.  We have two generating objects, $+$ and $-$, the two signed points.  A TQFT must assign these objects vector spaces, which we’ll call $V$ and $W$.  Collections of points get assigned tensor products of all the corresponding vector spaces, since the functor is monoidal, so knowing these two vector spaces determines what $Z$ does to all objects.

What does $Z$ do to morphisms?  Well, some generating morphsims of interest are cups and caps: these are lines which connect a positive to a negative point, but thought of as cobordisms taking two points to the empty set, and vice versa.  That is, we have an evaluation:This statement is what is generalized to say that $n$-dimensional TQFT’s are classified by “fully” dualizable objects.

$ev: W \otimes V \rightarrow \mathbb{C}$

and a coevaluation:

$coev: \mathbb{C} \rightarrow V \otimes W$

Now, since cobordisms are taken up to equivalence, which in particular includes topological deformations, we get a bunch of relations which these have to satisfy.  The essential one is the “zig-zag” identity, reflecting the fact that a bent line can be straightened out, and we have the same 1-morphism in $Born^{fr}_1(1)$.  This implies that:

$(ev \otimes id) \circ (id \otimes coev) : W \rightarrow W \otimes V \otimes W \rightarrow W$

is the same as the identity.  This in turn means that the evaluation and coevaluation maps define a nondegenerate pairing between $V$ and $W$.  The fact that this exists means two things.  First, $W$ is the dual of $V$: $W \cong V*$.  Second, this only makes sense if both $V$ and its dual are finite dimensional (since the evaluation will just be the trace map, which is not even defined on the identity if $V$ is infinite dimensional).

On the other hand, once we know, $V$, this determines $W \cong V*$ up to isomorphism, as well as the evaluation and coevaluation maps.  In fact, this turns out to be enough to specify $Z$ entirely.  The classification then is: 1-D TQFT’s are classified by finite-dimensional vector spaces $V$.  Crucially, what made finiteness important is the existence of the dual $V*$ and the (co)evaluation maps which express the duality.

In an $(\infty,n)$-category, to say that an object is “fully dualizable” means more that the object has a dual (which, itself, implies the existence of the morphisms $ev$ and $coev$). It also means that $ev$ and $coev$ have duals themselves – or rather, since we’re talking about morphisms, “adjoints”. This in turn implies the existence of 2-morphisms which are the unit and counit of the adjunctions (the defining properties are essentially the same as those for morphisms which define a dual). In fact, every time we get a morphism of degree less than $n$ in this process, “fully dualizable” means that it too must have a dual (i.e. an adjoint).

This does run out eventually, though, since we only require this goes up to dimension $(n-1)$: the $n$-morphisms which this forces to exist (quite a few) aren’t required to have duals. This is good, because if they were, since all the higher morphisms available are invertible, this would mean that the dual $n$-morphisms would actually be weak inverses (that is, their composite is isomorphic to the identity)… But that would mean that the dual $(n-1)$-morphisms which forced them to exist would also be weak inverses (their composite would be weakly isomorphic to the identity)… and so on! In fact, if the property of “having duals” didn’t stop, then everything would be weakly invertible: we’d actually have a (weak) $\infty$-groupoid!

### Classifying TQFT

So finally, the point of the Cobordism Hypothesis is that a (fully extended) TQFT is a functor $Z$ out of this $nBord^{fr}_n(n)$ into some target $(\infty,1)$-category $\mathcal{C}$. There are various options, but whatever we pick, the functor must assign something in $\mathcal{C}$ to the point, say $Z(pt)$, and something to each of $ev$ and $coev$, as well as all the higher morphisms which must exist. Then functoriality means that all these images have to again satisfy the properties which make $Z(pt)$ a fully dualizable object. Furthermore, since $nBord^{fr}_n(n)$ is the free gadget with all these properties on the single object $pt$, this is exactly what it means that $Z$ is a functor. Saying that $Z(pt)$ is fully dualizable, by implication, includes all the choices of morphisms like $Z(ev)$ etc. which show it as fully dualizable. (Conceivably one could make the same object fully dualizable in more than one way – these would be different functors).

So an extended $n$-dimensional TQFT is exactly the choice of a fully dualizable object $Z(pt) \in \mathcal{C}$, for some $(\infty,n)$-category $\mathcal{C}$. This object is “what the TQFT assigns to a point”, but if we understand the structure of the object as a fully dualizable object, then we know what the TQFT assigns to any other manifold of any dimension up to $n$, the highest dimension in the theory. This is how this algebraic characterization of cobordisms helps to classify such theories.

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.

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.

Among the talks given in our seminar on stacks and groupoids, there have been a few which I haven’t posted about yet – two by Tom Prince about stacks and homotopy theory, and one by José Malagon-Lopez comparing different characterizations of stacks. Tom is a grad student, and José is a postdoc, and they both work with Rick Jardine, who has done a lot of important work in homotopy theory, notably from the simplicial point of view. There was some overlap, since José was comparing the different characterizations for stacks that had been used by different people through the seminar, including Tom, but there’s still quite a lot to say here. I’ll try to cover the main points as I understand them, focusing on what I personally find relevant.

A major theme for both of them is the use of descent, which in general is a way to talk about the objects of a category in terms of another category. A standard example of descent would be the case of sheaves. First, though, what is it that’s being described in terms of descent?

Well, there are two opposite points of view on stacks – as categories fibred in groupoids (CFG’s), and as sheaves of groupoids. (I’ve found this book by Behrend et al. on algebraic stacks handy in parsing through some of the definitions here, and Jose recommended Vistoli’s notes on sites, fibred categories, and descent) One of the things Jose summarized in his talk was how these are related (which was a key bit of Aji’s earlier talk, blogged here). A CFG over $\mathcal{S}$ is a functor $p: \mathcal{X} \rightarrow \mathcal{S}$ where the preimage over $(x,1_x)$ is a groupoid (that is, all the morphisms mapping to an identity are invertible).

Now, given such $p : \mathcal{X} \rightarrow \mathcal{S}$ one gets a (weak) functor from $\mathcal{S}$ into groupoids (the “fibre-selecting” functor, which, among other conditions, gives the groupoid $p^{-1}(x,1_x)$ for each object $x$. Specifying this and showing it is a weak functor takes a little work. But in particular, there are properties on CFG’s a stack is such a functor into $Gpd$ with the extra property that descent data are effective. This is a weak version of the condition for a sheaf.

Stacks and Descent

The classical setting for descent questions is sheaf theory. To begin with, we have some category $\mathcal{S}$ of spaces – this might be $Top$ (topological spaces), or $Sch$ (affine schemes), or something else – the classical version has $\mathcal{S} = \mathcal{O}(X)$, the category of open sets on a topological space. The main thing is that $\mathcal{S}$ must be a Grothendieck site; in particular, there is a notion of covering for an object $X \in \mathcal{S}$. This is a collection $\underline{U} = \{ f_{\alpha} : U_{\alpha} \rightarrow X \}$ of arrows satisfying some conditions that capture the intuitive idea of “open cover”.

So, just to recall: the idea of describing a space as a sheaf on a site involves a little shift of perspective, but it’s the idea behind diffeological spaces (as I described in my post on Enxin Wu’s talk in our seminar, and which, for me, is a good example to help understand this viewpoint). A diffeological space is determined by giving the set of all “smooth” maps into it from each object in a certain site. Now, any space $S \in \mathcal{S}$ can also be represented in $Hom(\mathcal{S}^{op},Set)$ (by the Yoneda embedding) as the sheaf $Hom(-,S)$ which gives, for each space $X$, the set of maps in $\mathcal{S}$ (topological, algebraic, or whatever) into $S$ – but one can get objects in a bigger category, namely that of sheaves, which is a way of describing them in terms of the objects in the site $\mathcal{S}$. In the case of diffeological spaces, the site in question is just the one consisting of neighborhoods in $\mathbb{R}^n$ for any $n$, with smooth maps, and the obvious idea of a cover. So representable ones are just Euclidean neighborhoods, and general ones are defined by smooth maps out of these: the sheaf condition is just a way to state the natural compatibility condition for these maps. Similar thinking applies to any site $\mathcal{S}$.

The point of this condition is to ask when we can take a cover of an object $S$, and describe global objects (functions on $S$) in terms of local objects (functions on elements in the cover), which are compatible. Descent is the gluing condition for a sheaf $F$: given a cover – a bunch of maps $f_i : U_i \rightarrow S$ which satisfy some conditions that capture the intuitive idea of covering $S$ – a descent datum is a collection of $x_i \in F(U_i)$, and isomorphisms between the restrictions (by $F(\leq)$) to overlaps $U_i \cap U_j$, where the isomorphisms satisfy some cocycle condition ensuring that restrictions to $U_i \cap U_j \cap U_k$ are equal. The datum is effective if all there is a “global” object $x \in F(S)$ where $x_i$ is the restriction of $x$. (I find this easiest to see when $\mathcal{S}=\mathcal{O}(X)$, where it says we can glue functions on local patches that agree on overlaps, and find that they must have come by restricting a global function on $X$.)

This all makes sense if $F$ has values in $Set$ (or some other 1-category), but the point for stacks is that we have a weak functor $G : \mathcal{S}^{op} \rightarrow Gpd$. That is, the values are in groupoids, which naturally form a 2-category. So the descent can be weakened – instead of an equality in the cocycle condition, we get an isomorphism, which has to be coherent. Part of the point of describing stacks as “sheaves of groupoids” is as a weakening this way of describing a space, to an “up to equivalence” kind of condition.

One point which Jose made, and which Tom made use of, is that this description of a Grothendieck topology really gives too much information – that is, the category of sheaves on a site (taken up to equivalence) doesn’t uniquely determine the site. Instead of coverings, one should talk about sieves – these are, one might say, one-sided ideals of maps into $S$. In particular, subfunctors $R \subset Hom(-,S)$ – that is, for each space $V$, a subset of all maps $V \rightarrow S$, in a way that gets along with composition of maps (which is how they resemble ideals). Any covering defines a seive – as the subfunctor of maps which factor through the covering maps – but more than one covering might define the same sieve (rather the same way an ideal can be presented in terms of different generators).

So the view of stacks as sheaves $G$ (of groupoids) satisfying descent is then rephrased by saying that, for any covering sieve $R$ of an object $S \in \mathcal{S}$, there is an equivalence of functors between $Hom(E_S, G)$ and $Hom(E_R,G)$, where $E_S$ and $E_R$ are some sheaves on $\mathcal{S}$ constructed in a fairly natural way from the object $S$ itself, and from the sieve $R$. The point is that $Hom(E_S,G) = G(S)$ is a groupoid. The functor $E_R$ ends up such that $Hom(E_R,G)$ can be described in terms of covers $\{ U_i \rightarrow S \}$ as having objects which are compatible collections of objects from $U_i$ and isomorphisms between their restrictions – that is, descent data – and morphisms being compatible maps. So equivalence of these (2-)functors ends up being the stack condition.

One of Tom’s objectives was to look at all this from the point of view of simplicial sheaves – and here we need to think about homotopy-theoretic ideas of “equivalence”, instead of just the equivalence of categories we just used.

Model Structure

One of the major tools in homotopical algebra is the notion of a model structure (these slides by Peter May give the basic concepts). These show up throughout higher category theory because homotopies-between-homotopies-…-between-maps give a natural model of higher morphisms.

Model categories axiomatize three special kinds of maps one is interested in when talking about maps between spaces, up to homotopy. “Weak equivalence” generalizes a “homotopy equivalence” $f : X \rightarrow Y$ – a map which induces isomorphisms between homotopy groups of $X$ and $Y$ (as far as homotopy theory can detect, $X$ and $Y$ are “the same”). “Fibration” and “cofibration” are defined in homotopy theory by a lifting property (and its dual) – essentially, that if a map can be lifted along $f$, so can a homotopy of the map.  Fibrations generalize (“nice”) surjections, and cofibrations generalize (“nice”) inclusions.

In particular, Tom was making use of a notion of descent where the equations that define the descent conditions are just required to be weak equivalences. The point is that we can talk about sheaves of various kinds of things – sets, groupoids, or simplicial sets were the examples he gave. The relevant notion of equivalence for sets is isomorphism (the usual way of stating descent), but for groupoids it’s equivalence, and for simplicial sets, it’s another notion of weak equivalence (from the Joyal-Tierney model structure). When talking about stacks, we’re dealing with groupoids.

On the other hand, groupoids can be described in terms of simplicial sets, using the construction known as the simplicial nerve. In particular the classifying spaces of groupoids have no interesting homotopy groups above the first – so this ends up giving another way to state the weakened form of descent mentioned above. This type of construction – using the fact that simplicial sets are very versatile (can describe categories, or reasonable spaces, one $\infty$-categories, for instance), is what makes the study of simplicial presheaves, which is the basis of a lot of work by Rick Jardine (see the book Simplicial Homotopy Theory for a whole lot more that I can touch on here).

This gives another characterization of stacks: a sheaf of groupoids $G$ is a stack if and only if $BG$ (sheaf of classfying spaces), satisfies descent in that it is “pointwise” (that is, section-wise) weakly-equivalent to a certain kind of “globally fibrant replacement”. This is like the description of descent in terms of an equivalence of categories, as above – but in general is weaker. In fact, when the simplicial sets we’re talking about are classifying spaces for groupoids, then by construction these are just the same. This kind of replacement accomplishes for stacks roughly what “sheafification” does for sheaves – i.e. turns “prestacks” into “stacks”. This is done by taking a limit over all sieves – the universal property of the limit, then, is what ensures the existence of all the global objects that descent requires must exist. This is always a “local” weak equivalence, but only if we started with a stack is it one “pointwise” (i.e. in terms of sections).

Cocycles

As an aside: one thing which Tom talked about as a preliminary, but which I found particularly helpful from where I was coming from, had to do with “cocycle categories”. This is a somewhat unusual use of the term “cocycle”: here, a cocycle from $X$ to $Y$ is a certain kind of span – namely, a pair of maps from $Z$:

$X \stackrel{f}{\leftarrow} Z \stackrel{g}{\rightarrow} Y$

where $f$ is a “weak equivalence”. A morphism between cocycles is just a map $Z \rightarrow Z'$ which commutes with those in the cocycle. These form a category $H(X,Y)$. The point of introducing this is to say that there is a correspondence between components in this category – that is, $\pi_0(H(X,Y))$ and homotopy classes of maps from $X$ to $Y$ (the collection of which is denoted $[X,Y]$ in homotopy theory).

One way to think about this is that cocycles stand in relation to functions, roughly, as spans stand to relations. If we are in $Sets$, where weak equivalence is isomorphism, then $Z$ can be thought of as the graph of a function from $X$ to $Y$ – since $f$ is bijective, $Z$ can stand as a substitute for $X$. Moving to spaces, we weaken the requirement so that $Z$ is only a replacement for $X$ “up to homotopy” – thus, cocycles are adequate replacements for homotopy classes of functions. This business of replacing objects with other, nicer objects (say, “fibrant replacement”) is a recurring theme in homotopy theory. This digression on cocycles helped me understand why. Part of the point is that the equivalence classes of these “cocycles” is easier to calculate directly than, but equivalent to, homotopy classes of maps.

In any case, there’s more I could say about these talks, but I’ll leave off for now.

Over the next week, I’ll be visiting Derek Wise at UC Davis, to talk about some stuff having to do with ETQFT’s , but soon enough I’ll also do a writeup of Emre Coskun’s talks in the seminar about gerbes, which started today and continue tomorrow.

It’s taken me a while to write this up, since I’ve been in the process of moving house – packing and unpacking and all the rest. However, a bit over a week ago, I was in Montreal, attending MakkaiFest ’09 at the Centre de Recherches Mathematiques at the University of Montréal (and a pre-conference workshop hosted at McGill, which I’m including in the talks I mention here). This was in honour of the 70th birthday of Mihaly (Michael) Makkai, of McGill University. Makkai has done a lot of important foundational work in logic, model theory, and category theory, and a great many of the talks were from former students who’d gone on and been inspired by him, so one got sense of the range of things he’s worked on through his life.

The broad picture of Makkai’s work was explained to us by J.P. Marquis, from the Philosophy department at U of M. He is interested in philosophy of mathematics, and described Makkai’s project by contrast with the program of axiomatization of the early 20th century, along the lines suggested by Hilbert. This program provided a formal language for concrete structures – the problem, which category theory is part of a solution to, is to do the same for abstract structures. Contrast, for instance, the concrete description of a group $G$ as a (particular) set with some (particular) operation, with the abstract definition of a group object in a category. Makkai’s work in categorical logic, said Marquis, is about formalizing the process of abstraction that example illustrates.

Model Theory/Logic

This matter – of the relation between abstract theories and concrete models of the theories – is really what model theory is about, and this is one of the major areas Makkai has worked on. Roughly, a theory is most basically a schema with symbols for types, members of types, and some function symbols – and a collection of sentences built using these symbols (usually generated from some axioms by rules of logical inference). A model is (intuitively), an interpretation of the terms: a way of assigning concrete data to the symbols – say, a symbol for a type is assigned the set of all entities of that type, and a function symbol is assigned an actual function between sets, and so on – making all propositions true. A morphism of models is a map that preserves all the properties of the model that can be stated using first order logic.

This is an older way to say things – Victor Harnik gave an expository talk called “Model Theory vs. Categorical Logic” in which he compared two ways of adding an equivalence relation to a theory. The model theory way (invented by Shelah) involves taking the theory (list of sentences) $T$ and extending it to a new theory $T^{eq}$. This has, for instance, some new types – if we had a type for “element of group”, for example, we might then get a new type “equivalence class of elements of group”, and so on. Now, this extension is “tight” in the sense that the categories of all models of $T$ and of $T^{eq}$ are equivalent (by a forgetful functor $Mod(T^{eq}) \rightarrow Mod(T)$) – but one can prove new theorems in the extended theory. To make this clear, he described work (due to Makkai and Reyes) about pretopos completion. Here, one has the concept of a “Boolean logical category” – $Set$ is an example, as is, for any theory, a certain category whose objects are the formulas of the theory. This is related to Lawvere theories (see below). There are logical functors between such categories – functors into $Set$ are models, but there are also logical functors between theories. The point is that a theory $T$ embeds into $T^{eq}$ (abusing notation here – these are now the boolean logical categories). Then the point is that $T^{eq}$ arises as a kind of completion of $T$ – namely, it’s a boolean pretopos (not just category). Moreover, it has some nice universal properties, making this point of view a bit more natural than the model-theoretic construction.

Bradd Hart’s talk, “Conceptual Completeness for Cantinuous Logic”, was a bit over my head, but made some use of this kind of extension of a theory to $T^{eq}$. The basic point seems to be to add some kind of continuous structure to logic. One example comes from a metric structure – defining a metric space of terms, where the metric function $d(x,y)$ is some sum $\sum_n \phi_n (x,y)$, where the $\phi_n$ are formulas with two variables, either true or false – where true gives a $0$, and false gives a $1$ in this sum. This defines a distance from $x$ to $y$ associated to the given list of formulas $\phi_n$. A continuous logic is one with a structure like this. The business about equivalence relations arises if we say two things are equivalent when the distance between them is 0 – this leads to a concept of completion, and again there’s a notion that the categories of models are equivalent (though proving it here involves some notion of approximating terms to arbitrary epsilon, which doesn’t appear in standard logic).

Anand Pillay gave a talk which used model theory to describe some properties of the free group on n generators. This involved a “theory of the free group” which applies to any free group, and regard each such group as a model of the theory – in fact a submodel of some large model, and using model-theoretic methods to examine “stability” properties, in some sense which amounts to a notion of defining “generic” subsets of the group.

Logic and Higher Categories

A number of talks specifically addressed the ground where logic meets higher dimensional categories, since Makkai has worked with both.

In one talk, Robert Paré described a way of thinking about first-order theories as examples of “double Lawvere theories”. Lawvere’s way of formalizing “theories and models” was to say that the theory is a category itself (which has just the objects needed to describe the kind of structure it’s a theory of) – and a model is a functor into $Sets$ (or some other category – a model of the theory of groups in topological spaces, say, is a topological group). For example, the theory of groups includes an object $G$ and powers of it, multiplication and inverse maps, and expresses the axioms by the fact that certain diagrams commute. A model is a functor $M : Th(Grp) \rightarrow Sets$, assigning to the “group object” a set of elements, which then get the group structure from the maps. Instead of a category, this uses a double category. There are two kinds of morphisms – horizontal and vertical – and these are used to represent two kinds of symbols: function symbols, and relation symbols. (For example, one can talk about the theory of an ordered field – so one needs symbols for multiplication and addition and so forth, but also for the order relation $\leq$). Then a model of such a theory is a double functor into the double category whose objects are sets, and whose horizontal and vertical morphisms are respectively functions and relations.

André Joyal gave a talk about the first order logic of higher structures. He started by commenting on some fields which began life close together, and are now gradually re-merging: logic and category theory; category theory and homotopy theory (via higher categories); homotopy theory and algebraic geometry. The higher categories Joyal was thinking of are quasicategories, or “$( \infty, 1)$-categories, which are simplicial sets satisfying a weak version of a horn-filling condition (the “strict” version of this, a Kan complex, includes as example $N(C)$, the nerve of a category $C$ – there’s an n-simplex for each sequence of n composable morphisms, whose other edges are the various composites, and whose faces are “compositors”, “associators”, and so on – which for $N(C)$ are identities). The point of this is that one can reproduce most of category theory for quasicategories – in particular, he mentioned limits and colimits, factorization systems, pretoposes, and model theory.

Moving to quasicategories on one side of the parallel between category theory and logic has a corresponding move on the other side – on the logic side, one aspect is that the usual notion of a language is replaced by what’s called Martin-Löf type theory. This, in fact, was the subject of Michael Warren’s talk, “Martin-Löf complexes” (I reported on a similar talk he gave at Octoberfest last year). The idea here is to start by defining a globular set, given a theory and type $A$ – a complex whose n-cells have two faces, of dimension (n-1). The 0-cells are just terms of some type $A$. The 1-cells are terms of types like $\underline{A}(a,b)$, where $a$ and $b$ are variables of type $A$ – the type has an interpretation as a proposition that $a=b$ “extensionally” (i.e. not via a proof – but as for instance when two programs with non-equivalent code happen to always produce the same output). This kind of operation can be repeated to give higher cells, like $\underline{A(a,b)}(f,g)$, and so on. Given a globular set $G$, one gets a theory by an adjoint construction. Putting the two together, one has a monad on the category of globular sets – algebras for the monad are Martin-Löf complexes. Throwing in syntactic rules to truncate higher cells (I suppose by declaring all cells to be identities) gives n-truncated versions of these complexes, $MLC_n$. Then there is some interesting homotopy theory, in that the category of n-truncated Martin-Löf complexes is expected to be a model for homotopy n-types. For example, $MLC_0$ is equivalent to $Sets$, and there is an adjunction (in fact, a Quillen equivalence – that is, a kind of “homotopy” equivalence) between $MLC_1$ and $Gpd$.

Category Theory/Higher Categories

There were a number of talks that just dealt with categories – including higher categories – in their own right. Makkai has worked, for example, on computads, which were touched on by Marek Zawadowski in one of his two talks (one in the pre-conference workshop, the other in the conference). The first was about categories of “many-to-one shapes”, which are important to computads – these are a notion of higher-category, where every cell takes many “input” faces to one “output” face. Zawadowski described a “shape” of an n-cell as an initial object in a certain category built from the category of computads with specified faces. Then there’s a category of shapes, and an abstract description of “shape” in terms of a graded tensor theory (graded for dimension, and tensor because there’s a notion of composition, I believe). Zawadowski’s second talk, “Opetopic Sets in Lax Monoidal Fibrations”, dealt with a similar topic from a different point of view. A lax monoidal fibration (LMF) is a kind of gadget for dealing with multi-level structures (categories, multicategories, quasicategories, etc). There’s a lot of stuff here I didn’t entirely follow, but just to illustrate: categories arise as LMF, by the fibration $cod : Set^{B} \rightarrow Set$, where $B$ is the category with two objects $M, O$, and two arrows from $M$ to $O$. An object in the functor category $Set^{B}$ consists of a “set of morphisms and set of objects” with maps – making this a category involves the monoidal structure, and how composition is defined, and the real point is that this is quite general machinery.

Joachim Lambek and Gonzalo Reyez, both longtime collaborators and friends of Makkai, also both gave talks that touched on physics and categories, though in very different ways. Lambek talked about the “Lorentz category” and its appearance in special relativity.  This involves a reformulation of SR in terms of biquaternions: like complex numbers, these are of the form $u + iv$, but $u$ and $v$ are quaternions.  They have various conjugation operations, and the geometry of SR can be described in terms of their algebra (just as, say, rotations in 3D can be described in terms of quaternions).  The Lorentz category is a way of organizing this – its two objects correspond to “unconjugated” and “conjugated” states.

Gonzalo Reyez gave a derivation of General Relativity in the context of synthetic differential geometry.  The substance of this derivation is not so different from the usual one, but with one exception.  Einstein’s field equations can be derived in terms of the motions of small regions full of of freely falling test particles – synthetic differential geometry makes it possible to do the same analysis using infinitesimals rigorously all the way through.  The basic point here is that in SDG one replaces the real line as usually conceived, with a “real line with infinitesimals” (think of the ring $\mathbb{R}[\epsilon]/\langle \epsilon^2 \rangle$, which is like the reals, but has the infinitesimal $\epsilon$, whose square is zero).

Among other talks: John Power talked about the correspondence between Lawvere theories in universal algebra and finitary tree monads on sets – and asked about what happens to the left hand side of this correspondence when we replace “sets” with other categories on the righ hand side. Jeff Egger talked about measure theory from a categorical point of view – namely, the correspondence of NCG between C*-algebras and “noncommutative” topological spaces, and between W*-algebras and “noncommutative” measure spaces, thought of in terms of locales. Hongde Hu talked about the “codensity theorem”, and a way to classify certain kinds of categories – he commented on how it was inspired by Makkai’s approach to mathematics: 1) Find new proofs of old theorems, (2) standardize the concepts used in them, and (3) prove new theorems with those concepts. Fred Linton gave a talk describing Heath’s “V-space”, which is a half-plane with a funny topology whose open sets are “V” shapes, and described how the topos of locally finite sheaves over it has surprising properties having to do with nonexistence of global sections. Manoush Sadrzadeh, whom I met recently at CQC (see the bottom of the previous post) was again talking about linguistics using monoidal categories – she described some rules for “clitic movement” and changes in word order, and what these rules look like in categorical terms.

Other

A few other talks are a little harder for me to fit into the broad classification above.  There was Charles Steinhorn’s talk about ordered “o-minimal” structures, which touched on a bit of economics – essentially, a lot of economics is based on the assumption that preference orders can be made into real-valued functions, but in fact in many cases one has (variants on) “lexicographic order”, involving ranked priorities.  He talked about how typically one has a space of possibilities which can be cut up into cells, with one sort of order in each cell.  There was Julia Knight, talking about computable structures of “high Scott rank” – in particular, this is about infinite structures that can still be dealt with computably – for example, infinitary logical formulas involving an infinite number of “OR” statements where all the terms being joined are of some common form.  This ends up with an analysis of certain infinite trees.  Hal Kierstead gave a talk about Ramsey theory which I found notable because it used the kind of construction based on a game: to prove that any colouring of a graph (or hypergraph) has some property, one devises a game where one player tries to build a graph, and the other tries to colour it, and proves a winning strategy for one player.  Finally, Michael Barr gave a talk about a duality between certain categories of modules over commutative rings.

All in all, an interesting conference, with plenty of food for thought.

In the last couple of weeks of the winter term, there were two series of talks here at UWO, by different speakers, from very different points of view, which bear on the subject of moduli spaces of connections.

There seem to be several schools of thought approaching the subject of moduli spaces, and in particular how to handle the reduction by symmetries without losing too much – three approaches I know of are the symplectic point of view (thinking of the moduli space as a symplectic space, or perhaps orbifold, and reduction by taking whole “leaves” to points), the algebraic-geometric (describing them using Deligne-Mumford stacks), and the groupoid point of view (which is the one I’m most familiar with). I suppose, in light of my previous note, that there must be a noncommutative-geometry view of the subject, though if anyone is using NCG to look at these moduli spaces in particular I don’t know who. Before talking about reducing the moduli spaces, there’s already a lot to say about them which people have studied in some detail.

The first speaker here who touched on this was Fred Cohen, who gave a series of three talks about special subspaces of products (and talked a lot about about stable homotopy theory). The second was Eduardo Gonzalez, who gave a seminar and a colloquium talk on equivariant Gromov-Witten theory. I’ll try to briefly give an overview of what they each had to say, mainly focusing on this common element.

Part 1 – Talks by Fred Cohen

Fred Cohen was speaking about various subspaces of products. He was summarizing a number of different projects, including for example this (on loop spaces of configuration spaces) and this (about spaces of homomorphisms). The first talk dealt with the seemingly simple space $Conf(X,n) = \{(x_1, \dots, x_n) | x_i \neq x_j \text{ when } i \neq j\}$ of distinct n-tuples of points in a space $X$, and the related natural space $Conf(X,n)/S_n$ (the action of the symmetric group makes the points unlabeled). In the case $X= \mathbb{R}^2$, a point in $Conf(\mathbb{R}^2,n)$ is a list of n distinct points. So a loop in this space is a motion of the n points which returns them to their original locations – considered up to homotopy, this is just a braid. In fact, $\pi_1(Conf(\mathbb{R}^2,n)) = PB_n$, the n-strand pure braid group; and $\pi_1(Conf(\mathbb{R}^2,n)/S_n) = B_n$, the full braid group (points needn’t end up in their original positions). In fact, the configuration spaces are $K(\pi,1)$ spaces – that is, they are classifying spaces of these groups, and have no higher homotopy groups above $\pi_1$.

Replacing $X = \mathbb{R}^2$ here with $X=S$, a surface, the same sort of thing defines the n-strand “surface braid group” for $S$, which is $P_n(S) = \pi_1(Conf(S,n))$. We heard how this decomposes in terms of the “Borromean” braid group – the subgroup of braids which become disconnected when you remove one strand (this is the kernel of a map induced by the projections into $Conf(S,n-1)$).

There was more about the homotopy type of these spaces, and a second talk covered “moment-angle complexes”, but here I’m interested in Cohen’s third talk about subspaces of products. This was on “representations” of a discrete group, which in this context means – almost – homomorphisms into a chosen group $G$. (If $G = GL(n)$, these are the more famous linear representations.) This is related to subspaces of the product $G^N$, which arise from looking at the moduli space $Hom(\pi,G)$, where $\pi$ is a discrete group and $G$ a topological group.

In particular, if $\pi = \pi_1(X)$, for a space $X$, such a representation can be thought of as a $G$-connection. In this picture, a connection is just a way of assigning an element of the gauge group $G$ to each path in $X$.) Actually, I mentioned this is “almost” the space of representations, which is actually $Rep(\pi,G) = Hom(\pi,G)/G$ – the moduli space of flat connections modulo gauge transformations. A gauge transformation (assuming $X$ is connected) acts by conjugation: $g(\gamma) \rightarrow h g(\gamma) h$, for a class $\gamma$ of loops in $X$.

This is the usual way of looking at this moduli space of geometric structures – I’ve mentioned here the alternative view that a flat connection is a functor $g : \Pi_1(M) \rightarrow$, and a gauge transformation is a natural transformation. Then the moduli space becomes a moduli stack, which as mentioned above I tend to think of as a groupoid. But the moduli spaces of homomorphisms (the objects) and representations (isomorphism classes of objects) carry a lot of information. Particular cases which Cohen discussed were $\pi = F_n$, the free group on $n$ generators, and $\mathbb{Z}^n$, the free abelian group on $n$ generators. These are fundamental groups of, respectively, the $n$-punctured plane and the genus-$n$ torus. Now $Hom(F_n,G) \cong G^n$, and the map $F_n \rightarrow \mathbb{Z}^n$ induces an inclusion $Hom(\mathbb{Z}^n,G) \stackrel{i}{\rightarrow} Hom(F_n,G)$ – in fact it’s a subvariety – so this is a subset of a product, and techniques for dealing with these were Cohen’s real subject.

One that he discussed (described in the paper linked above by Adem, Cohen and Torres-Giese) uses the “descending central series” of $F_n$. This is a sequence of subgroups $\Gamma^q$ generated by the $q$-fold commutators $[\dots[g_1,g_2],g_3],\dots, g_q]$. In particular, one looks at the groups $F_n/\Gamma^q$, and in fact their spaces of homomorphisms:

$Hom(F_n/\Gamma^2,G) \subset Hom(F_n/\Gamma^3,G) \subset \dots \subset G^n$

So there’s a filtration of spaces associated to $F_n$ and $G$.

Now it’s pretty standard that there are maps $d_i : Hom(F_n,G) \rightarrow Hom(F_{n-1},G)$ (by dropping the $i^{th}$ generator), and $s_j : Hom(F_n,G) \rightarrow Hom(F_{n+1},G)$ (sending the extra generator to the identity). These, thought of as face and degeneracy maps, turn the collection of spaces $G^n$ (for all $n$) into a simplicial space. This has a geometric realization, which is the classifying space $BG$ (or, shifting which set is considered to be the $n$-simplices, $EG$, where $BG = EG/G$, and there’s a bundle $EG \rightarrow BG$). BUT, each of the $Hom(F_n,G)$ has the filtration above – so it turns out there’s a filtration of simplicial spaces, and in fact of bundles. The paper above uses this to find the cohomology, fundamental group, and so on of the spaces I just mentioned – including the moduli space of connections.

(Then Cohen talked about a generalization of this to arbitrary “transitively commutative” groups, but that takes us away from the geometry I started off talking about).

Part 2 – Talks by Eduardo Gonzalez

The second set of talks which touched on moduli spaces of connections was by Eduardo Gonzalez, related to stuff in this paper by Gonzalez and Chris Woodward speaking about gauged (or equivariant) Gromov-Witten invariants. These are discussed in this paper by Givental, and Gonzalez referenced several other people who’ve worked on related things, including Chen and Ruan (see this on GW theory for orbifolds), and Abramovich, Graber and Vistoli (see this, on GW theory for stacks). Strictly speaking, this doesn’t address just the moduli space of flat connections, but actually a more complex moduli space for a theory involving a choice of connection (on a bundle), and also a section of the bundle. It is called the moduli space of symplectic vortices, and is very much involved with symplectic geometry as you might expect.

The usual Gromov-Witten invariants, roughly, count the number of holomorphic curves on a $2k$-dimensional symplectic manifold $X$. (That is, $X$ has an exact symplectic form $\omega$ – i.e. $d \omega = 0$ and $\omega$ is nondegenerate – and there’s an almost-complex structure $J : TX \rightarrow TX$- that is $J^2 = -1$; these give a metric $g(u,v) = \omega(u, Jv)$). This $J$ determines a complex derivative $\partial_J$ in a natural way.

A curve is a map $u : \Sigma \rightarrow X$, where $\Sigma$ is a Riemann surface (i.e. complex curve), which is holomorphic if $\partial_J(u) = 0$. The moduli space $\mathcal{M}(\Sigma, X, J)$ of these holomorphic curves – which is also the space of sections of suitable bundles over $\Sigma$ – each one amounts to a choice of a particular bundle over $\Sigma$, and a connection and holomorphic section of the bundle. This is where the Gromov-Witten invariants come from. Actually, it comes from a compactification $M$ of the space of maps from $\Sigma$ with $n$ “marked” (distinguished) points (so here actually we start to circle back around to the configuration space $Conf(\Sigma, n)$ Fred Cohen talked about).

Given a cohomology class $\alpha \in H^2(X,\mathbb{Q})^n$ (that is, $n$ 2-cocycles), one gets a form which can be integrated over $M$. The Gromov-Witten invariant, for that choice of form, is just the total “volume” of the moduli space with respect to that form, $\int_M ev^(\alpha)$ (the form $\alpha$ is pulled back under the map evaluating it at the $n$ marked points). This is sometimes described (rather roughly) as “counting” the pseudoholomorphic maps.

One thing people seem to be quite interested in is how this is related to so-called “quantum cohomology” for the space $X$. Since the GW invariants take some forms and give numbers, the idea is that they can be used to define a “three point function” on cohomology classes (by taking all but three of the $n$ cocycles to be the fixed $\omega$), which in turn can be taken to be the structure coefficients for a deformation of the cup product for cohomology. (Take the cohomology ring, take its tensor product with a ring of power series, and write the new product as a power series whose first terms give the usual cup product).

However, what Gonzalez was talking about was “gauged” Gromov-Witten invariants, where spaces are replaced by stack – in particular, stacks that come from an action of a group $G$ on the space $X$ (which, since $X$ is a symplectic manifold, should preserve the form $\omega$). The symplectic geometry way to talk about this is one I’m not very familiar with, but Gonzalez referred to $X\/\!\!\/G$ as the “categorical quotient” (i.e. the transformation groupoid, in the language I’m more used to) or the “symplectic reduction” (here‘s a brief note on the subject, and here a long paper on the relevance to physics which I’m linking so I can find it later). Roughly, this is a two-step process, the second stage being a reduction to a quotient by a group action. The result, in general, will be a symplectic orbifold (if the action is free on orbits, it’ll be a manifold – otherwise, some orbits have extra symmetry, which give the special points of the orbifold).

In particular – and here we really get to the point of contact with the groupoid picture I’m more familiar with, the gauged GW invariants are associated to a space $M(P,X) = \mathcal{A}(P,X) \/\!\!\/ \mathcal{G}(P)$, where $\mathcal{A}(P,X)$ is a space of connections on some bundle $P \rightarrow X$, and $\mathcal{G}$ is the group of gauge transformations. Now, these aren’t the space of flat connections, which I’ve thought more about, but rather connections satisfying another equation, namely that the curvature plus a certain volume form should be zero (defining the volume form takes a while and I don’t get it in enough detail to try to sort it out here). Connections satisfying this equation are called vortices, for reasons which escape me.

But in any case, the invariants amount to some geometry-aware generalization of the groupoid cardinality of this orbifold, thought of as an (equivalence class of) groupoid(s), defined by the integral above. There is much more to say here, but it’s taken me long enough to write this up as is, so maybe I’ll return to those things in a separate post some time.

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