So it’s been a while since I last posted – the end of 2013 ended up being busy with a couple of visits to Jamie Vicary in Oxford, and Roger Picken in Lisbon. In the aftermath of the two trips, I did manage to get a major revision of this paper submitted to a journal, and put this one out in public. A couple of others will be coming down the pipeline this year as well.

I’m hoping to get back to a post about motives which I planned earlier, but for the moment, I’d like to write a little about the second paper, with Roger Picken.

Global and Local Symmetry

The upshot is that it’s about categorifying the concept of symmetry. More specifically, it’s about finding the analog in the world of categories for the interplay between global and local symmetry which occurs in the world of set-based structures (sets, topological spaces, vector spaces, etc.) This distinction is discussed in a nice way by Alan Weinstein in this article from the Notices of the AMS from

The global symmetry of an object X in some category \mathbf{C} can be described in terms of its group of automorphisms: all the ways the object can be transformed which leave it “the same”. This fits our understanding of “symmetry” when the morphisms can really be interpreted as transformations of some sort. So let’s suppose the object is a set with some structure, and the morphisms are set-maps that preserve the structure: for example, the objects could be sets of vertices and edges of a graph, so that morphisms are maps of the underlying data that preserve incidence relations. So a symmetry of an object is a way of transforming it into itself – and an invertible one at that – and these automorphisms naturally form a group Aut(X). More generally, we can talk about an action of a group G on an object X, which is a map \phi : G \rightarrow Aut(X).

“Local symmetry” is different, and it makes most sense in a context where the object X is a set – or at least, where it makes sense to talk about elements of X, so that X has an underlying set of some sort.

Actually, being a set-with-structure, in a lingo I associate with Jim Dolan, means that the forgetful functor U : \mathbf{C} \rightarrow \mathbf{Sets} is faithful: you can tell morphisms in \mathbf{C} (in particular, automorphisms of X) apart by looking at what they do to the underlying set. The intuition is that the morphisms of \mathbf{C} are exactly set maps which preserve the structure which U forgets about – or, conversely, that the structure on objects of \mathbf{C} is exactly that which is forgotten by U. Certainly, knowing only this information determines \mathbf{C} up to equivalence. In any case, suppose we have an object like this: then knowing about the symmetries of X amounts to knowing about a certain group action, namely the action of Aut(X), on the underlying set U(X).

From this point of view, symmetry is about group actions on sets. The way we represent local symmetry (following Weinstein’s discussion, above) is to encode it as a groupoid – a category whose morphisms are all invertible. There is a level-slip happening here, since X is now no longer seen as an object inside a category: it is the collection of all the objects of a groupoid. What makes this a representation of “local” symmetry is that each morphism now represents, not just a transformation of the whole object X, but a relationship under some specific symmetry between one element of X and another. If there is an isomorphism between x \in X and y \in X, then x and y are “symmetric” points under some transformation. As Weinstein’s article illustrates nicely, though, there is no assumption that the given transformation actually extends to the entire object X: it may be that only part of X has, for example, a reflection symmetry, but the symmetry doesn’t extend globally.

Transformation Groupoid

The “interplay” I alluded to above, between the global and local pictures of symmetry, is to build a “transformation groupoid” (or “action groupoid“) associated to a group G acting on a set X. The result is called X // G for short. Its morphisms consist of pairs such that  (g,x) : x \rightarrow (g \rhd x) is a morphism taking x to its image under the action of g \in G. The “local” symmetry view of X // G treats each of these symmetry relations between points as a distinct bit of data, but coming from a global symmetry – that is, a group action – means that the set of morphisms comes from the product G \times X.

Indeed, the “target” map in X // G from morphisms to objects is exactly a map G \times X \rightarrow X. It is not hard to show that this map is an action in another standard sense. Namely, if we have a real action \phi : G \rightarrow Hom(X,X), then this map is just \hat{\phi} : G \times X \rightarrow X, which moves one of the arguments to the left side. If \phi was a functor, then $\hat{\phi}$ satisfies the “action” condition, namely that the following square commutes:


(Here, m is the multiplication in G, and this is the familiar associativity-type axiom for a group action: acting by a product of two elements in G is the same as acting by each one successively.

So the starting point for the paper with Roger Picken was to categorify this. It’s useful, before doing that, to stop and think for a moment about what makes this possible.

First, as stated, this assumed that X either is a set, or has an underlying set by way of some faithful forgetful functor: that is, every morphism in Aut(X) corresponds to a unique set map from the elements of X to itself. We needed this to describe the groupoid X // G, whose objects are exactly the elements of X. The diagram above suggests a different way to think about this. The action diagram lives in the category \mathbf{Set}: we are thinking of G as a set together with some structure maps. X and the morphism \hat{\phi} must be in the same category, \mathbf{Set}, for this characterization to make sense.

So in fact, what matters is that the category X lived in was closed: that is, it is enriched in itself, so that for any objects X,Y, there is an object Hom(X,Y), the internal hom. In this case, it’s G = Hom(X,X) which appears in the diagram. Such an internal hom is supposed to be a dual to \mathbf{Set}‘s monoidal product (which happens to be the Cartesian product \times): this is exactly what lets us talk about \hat{\phi}.

So really, this construction of a transformation groupoid will work for any closed monoidal category \mathbf{C}, producing a groupoid in \mathbf{C}. It may be easier to understand in cases like \mathbf{C}=\mathbf{Top}, the category of topological spaces, where there is indeed a faithful underlying set functor. But although talking explicitly about elements of X was useful for intuitively seeing how X//G relates global and local symmetries, it played no particular role in the construction.

Categorify Everything

In the circles I run in, a popular hobby is to “categorify everything“: there are different versions, but what we mean here is to turn ideas expressed in the world of sets into ideas in the world of categories. (Technical aside: all the categories here are assumed to be small). In principle, this is harder than just reproducing all of the above in any old closed monoidal category: the “world” of categories is \mathbf{Cat}, which is a closed monoidal 2-category, which is a more complicated notion. This means that doing all the above “strictly” is a special case: all the equalities (like the commutativity of the action square) might in principle be replaced by (natural) isomorphisms, and a good categorification involves picking these to have good properties.

(In our paper, we left this to an appendix, because the strict special case is already interesting, and in any case there are “strictification” results, such as the fact that weak 2-groups are all equivalent to strict 2-groups, which mean that the weak case isn’t as much more general as it looks. For higher n-categories, this will fail – which is why we include the appendix to suggest how the pattern might continue).

Why is this interesting to us? Bumping up the “categorical level” appeals for different reasons, but the ones matter most to me have to do with taking low-dimensional (or -codimensional) structures, and finding analogous ones at higher (co)dimension. In our case, the starting point had to do with looking at the symmetries of “higher gauge theories” – which can be used to describe the transport of higher-dimensional surfaces in a background geometry, the way gauge theories can describe the transport of point particles. But I won’t ask you to understand that example right now, as long as you can accept that “what are the global/local symmetries of a category like?” is a possibly interesting question.

So let’s categorify the discussion about symmetry above… To begin with, we can just take our (closed monoidal) category to be \mathbf{Cat}, and follow the same construction above. So our first ingredient is a 2-group \mathcal{G}. As with groups, we can think of a 2-group either as a 2-category with just one object \star, or as a 1-category with some structure – a group object in \mathbf{Cat}, which we’ll call C(\mathcal{G}) if it comes from a given 2-group. (In our paper, we keep these distinct by using the term “categorical group” for the second. The group axioms amount to saying that we have a monoidal category (\mathcal{G}, \otimes, I). Its objects are the morphisms of the 2-group, and the composition becomes the monoidal product \otimes.)

(In fact, we often use a third equivalent definition, that of crossed modules of groups, but to avoid getting into that machinery here, I’ll be changing our notation a little.)

2-Group Actions

So, again, there are two ways to talk about an action of a 2-group on some category \mathbf{C}. One is to define an action as a 2-functor \Phi : \mathcal{G} \rightarrow \mathbf{Cat}. The object being acted on, \mathbf{C} \in \mathbf{Cat}, is the unique object \Phi(\star) – so that the 2-functor amounts to a monoidal functor from the categorical group C(\mathcal{G}) into Aut(\mathbf{C}). Notice that here we’re taking advantage of the fact that \mathbf{Cat} is closed, so that the hom-“sets” are actually categories, and the automorphisms of \mathbf{C} – invertible functors from \mathbf{C} to itself – form the objects of a monoidal category, and in fact a categorical group. What’s new, though, is that there are also 2-morphisms – natural transformations between these functors.

To begin with, then, we show that there is a map \hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}, which corresponds to the 2-functor \Phi, and satisfies an action axiom like the square above, with \otimes playing the role of group multiplication. (Again, remember that we’re only talking about the version where this square commutes strictly here – in an appendix of the paper, we talk about the weak version of all this.) This is an intuitive generalization of the situation for groups, but it is slightly more complicated.

The action \Phi directly gives three maps. First, functors \Phi(\gamma) : \mathbf{C} \rightarrow \mathbf{C} for each 2-group morphism \gamma – each of which consists of a function between objects of \mathbf{C}, together with a function between morphisms of \mathbf{C}. Second, natural transformations \Phi(\eta) : \Phi(\gamma) \rightarrow \Phi(\gamma ') for 2-morphisms \eta : \gamma \rightarrow \gamma' in the 2-group – each of which consists of a function from objects to morphisms of \mathbf{C}.

On the other hand, \hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C} is just a functor: it gives two maps, one taking pairs of objects to objects, the other doing the same for morphisms. Clearly, the map (\gamma,x) \mapsto x' is just given by x' = \Phi(\gamma)(x). The map taking pairs of morphisms (\eta,f) : (\gamma,x) \rightarrow (\gamma ', y) to morphisms of \mathbf{C} is less intuitively obvious. Since I already claimed \Phi and \hat{\Phi} are equivalent, it should be no surprise that we ought to be able to reconstruct the other two parts of \Phi from it as special cases. These are morphism-maps for the functors, (which give \Phi(\gamma)(f) or \Phi(\gamma ')(f)), and the natural transformation maps (which give \Phi(\eta)(x) or \Phi(\eta)(y)). In fact, there are only two sensible ways to combine these four bits of information, and the fact that \Phi(\eta) is natural means precisely that they’re the same, so:

\hat{\Phi}(\eta,f) = \Phi(\eta)(y) \circ \Phi(\gamma)(f) = \Phi(\gamma ')(f) \circ \Phi(\eta)(x)

Given the above, though, it’s not so hard to see that a 2-group action really involves two group actions: of the objects of \mathcal{G} on the objects of \mathbf{C}, and of the morphisms of \mathcal{G} on objects of \mathbf{C}. They fit together nicely because objects can be identified with their identity morphisms: furthermore, \Phi being a functor gives an action of \mathcal{G}-objects on \mathbf{C}-morphisms which fits in between them nicely.

But what of the transformation groupoid? What is the analog of the transformation groupoid, if we repeat its construction in \mathbf{Cat}?

The Transformation Double Category of a 2-Group Action

The answer is that a category (such as a groupoid) internal to \mathbf{Cat} is a double category. The compact way to describe it is as a “category in \mathbf{Cat}“, with a category of objects and a category of morphisms, each of which of course has objects and morphisms of its own. For the transformation double category, following the same construction as for sets, the object-category is just \mathbf{C}, and the morphism-category is \mathcal{G} \times \mathbf{C}, and the target functor is just the action map \hat{\Phi}. (The other structure maps that make this into a category in \mathbf{Cat} can similarly be worked out by following your nose).

This is fine, but the internal description tends to obscure an underlying symmetry in the idea of double categories, in which morphisms in the object-category and objects in the morphism-category can switch roles, and get a different description of “the same” double category, denoted the “transpose”.

A different approach considers these as two different types of morphism, “horizontal” and “vertical”: they are the morphisms of horizontal and vertical categories, built on the same set of objects (the objects of the object-category). The morphisms of the morphism-category are then called “squares”. This makes a convenient way to draw diagrams in the double category. Here’s a version of a diagram from our paper with the notation I’ve used here, showing what a square corresponding to a morphism (\chi,f) \in \mathcal{G} \times \mathbf{C} looks like:


The square (with the boxed label) has the dashed arrows at the top and bottom for its source and target horizontal morphisms (its images under the source and target functors: the argument above about naturality means they’re well-defined). The vertical arrows connecting them are the source and target vertical morphisms (its images under the source and target maps in the morphism-category).

Horizontal and Vertical Slices of \mathbf{C} // \mathcal{G}

So by construction, the horizontal category of these squares is just the object-category \mathbf{C}.  For the same reason, the squares and vertical morphisms, make up the category \mathcal{G} \times \mathbf{C}.

On the other hand, the vertical category has the same objects as \mathbf{C}, but different morphisms: it’s not hard to see that the vertical category is just the transformation groupoid for the action of the group of \mathbf{G}-objects on the set of \mathbf{C}-objects, Ob(\mathbf{C}) // Ob(\mathcal{G}). Meanwhile, the horizontal morphisms and squares make up the transformation groupoid Mor(\mathbf{C}) // Mor(\mathcal{G}). These are the object-category and morphism-category of the transpose of the double-category we started with.

We can take this further: if squares aren’t hip enough for you – or if you’re someone who’s happy with 2-categories but finds double categories unfamiliar – the horizontal and vertical categories can be extended to make horizontal and vertical bicategories. They have the same objects and morphisms, but we add new 2-cells which correspond to squares where the boundaries have identity morphisms in the direction we’re not interested in. These two turn out to feel quite different in style.

First, the horizontal bicategory extends \mathbf{C} by adding 2-morphisms to it, corresponding to morphisms of \mathcal{G}: roughly, it makes the morphisms of \mathbf{C} into the objects of a new transformation groupoid, based on the action of the group of automorphisms of the identity in \mathcal{G} (which ensures the square has identity edges on the sides.) This last point is the only constraint, and it’s not a very strong one since Aut(1_G) and G essentially determine the entire 2-group: the constraint only relates to the structure of \mathcal{G}.

The constraint for the vertical bicategory is different in flavour because it depends more on the action \Phi. Here we are extending a transformation groupoid, Ob(\mathbf{C}) // Ob(\mathcal{G}). But, for some actions, many morphisms in \mathcal{G} might just not show up at all. For 1-morphisms (\gamma, x), the only 2-morphisms which can appear are those taking \gamma to some \gamma ' which has the same effect on x as \gamma. So, for example, this will look very different if \Phi is free (so only automorphisms show up), or a trivial action (so that all morphisms appear).

In the paper, we look at these in the special case of an adjoint action of a 2-group, so you can look there if you’d like a more concrete example of this difference.

Speculative Remarks

The starting point for this was a project (which I talked about a year ago) to do with higher gauge theory – see the last part of the linked post for more detail. The point is that, in gauge theory, one deals with connections on bundles, and morphisms between them called gauge transformations. If one builds a groupoid out of these in a natural way, it turns out to result from the action of a big symmetry group of all gauge transformations on the moduli space of connections.

In higher gauge theory, one deals with connections on gerbes (or higher gerbes – a bundle is essentially a “0-gerbe”). There are now also (2-)morphisms between gauge transformations (and, in higher cases, this continues further), which Roger Picken and I have been calling “gauge modifications”. If we try to repeat the situation for gauge theory, we can construct a 2-groupoid out of these, which expresses this local symmetry. The thing which is different for gerbes (and will continue to get even more different if we move to n-gerbes and the corresponding (n+1)-groupoids) is that this is not the same type of object as a transformation double category.

Now, in our next paper (which this one was written to make possible) we show that the 2-groupoid is actually very intimately related to the transformation double category: that is, the local picture of symmetry for a higher gauge theory is, just as in the lower-dimensional situation, intimately related to a global symmetry of an entire moduli 2-space, i.e. a category. The reason this wasn’t obvious at first is that the moduli space which includes only connections is just the space of objects of this category: the point is that there are really two special kinds of gauge transformations. One should be thought of as the morphisms in the moduli 2-space, and the other as part of the symmetries of that 2-space. The intuition that comes from ordinary gauge theory overlooks this, because the phenomenon doesn’t occur there.

Physically-motivated theories are starting to use these higher-categorical concepts more and more, and symmetry is a crucial idea in physics. What I’ve sketched here is presumably only the start of a pattern in which “symmetry” extends to higher-categorical entities. When we get to 3-groups, our simplifying assumptions that use “strictification” results won’t even be available any more, so we would expect still further new phenomena to show up – but it seems plausible that the tight relation between global and local symmetry will still exist, but in a way that is more subtle, and refines the standard understanding we have of symmetry today.


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}))


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

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

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

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

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

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

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

Dold-Puppe Correspondence

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

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

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

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

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

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

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

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

Chain Homotopies and Quasi-Isomorphisms

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

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

given by the quotients

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

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

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

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

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

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

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

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

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

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

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

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

Derived Categories

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

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

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

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

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

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

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

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

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

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

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


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

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

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

For supermanifolds, the condition on the local rings says that, for fixed dimension (p \bar q ), a p|q-dimensional supermanifold has structure sheaf in which $they look like

\mathcal{O}(\mathcal{U}) \cong C^{\infty}(V,\mathbb{R}) \otimes \Lambda_q

In this, V is as above, and the notation

\Lambda_q = \Lambda ( \theta_1, \dots , \theta_q )

refers to the exterior algebra, which we can think of as polynomials in the \theta_i, with the wedge product, which satisfies \theta_i \wedge \theta_j = - \theta_j \wedge \theta_i. The idea is that one is supposed to think of this as the algebra of smooth functions on a space with p ordinary dimensions, and q “anti-commuting” dimensions with coordinates \theta_i. The commuting variables, say x_1,\dots,x_p, are called “bosonic” or “even”, and the anticommuting ones are “fermionic” or “odd”. (The term “fermionic” is related to the fact that, in quantum mechanics, when building a Hilbert space for a bunch of identical fermions, one takes the antisymmetric part of the tensor product of their individual Hilbert spaces, so that, for instance, v_1 \otimes v_2 = - v_2 \otimes v_1).

The structure sheaf picture can therefore be thought of as giving an atlas of charts, so that the neighborhoods locally look like “super-domains”, the super-geometry equivalent of open sets V \subset \mathbb{R}^p.

In fact, there’s a long-known theorem of Batchelor which says that any real supermanifold is given exactly by the algebra of “global sections”, which looks like \mathcal{O}(M) = C^{\infty}(M_{red},\mathbb{R}) \otimes \Lambda_q. That is, sections in the local rings (“functions on” open neighborhoods of M) always glue together to give a section in \mathcal{O}(M).

Another way to put this is that every supermanifold can be seen as just bundle of exterior algebras. That is, a bundle over a base manifold M_{red}, whose fibres are the “super-points” \mathbb{R}^{0|q} corresponding to \Lambda_q. The base space M_{red} is called the “reduced” manifold. Any such bundle gives back a supermanifold, where the algebras in the structure sheaf are the algebras of sections of the bundle.

One shouldn’t be too complacent about saying they are exactly the same, though: this correspondence isn’t functorial. That is, the maps between supermanifolds are not just bundle maps. (Also, Batchelor’s theorem works only for real, not for complex, supermanifolds, where only the local neighborhoods necessarily look like such bundles).

Why, by the way, say that \mathbb{R}^{0|q} is a super “point”, when \mathbb{R}^{p|0} is a whole vector space? Since the fermionic variables are anticommuting, no term can have more than one of each \theta_i, so this is a finite-dimensional algebra. This is unlike C{\infty}(V,\mathbb{R}), which suggests that the noncommutative directions are quite different. Any element of \Lambda_q is nilpotent, so if we think of a Taylor series for some function – a power series in the (x_1,\dots,x_p,\theta_1,\dots,\theta_q) – we see note that no term has a coefficient for \theta_i greater than 1, or of degree higher than q in all the \theta_i – so imagines that only infinitesimal behaviour in these directions exists at all. Thus, a supermanifold M is like an ordinary p-dimensional manifold M_{red}, built from the ordinary domains V, equipped with a bundle whose fibres are a sort of “infinitesimal fuzz” about each point of the “even part” of the supermanifold, described by the \Lambda_q.

But this intuition is a bit vague. We can sharpen it a bit using the functor of points approach…

Supermanifolds as Manifold-Valued Sheaves

As with schemes, there is also a point of view that sees supermanifolds as “ordinary” manifolds, constructed in the topos of sheaves over a certain site. The basic insight behind the picture of these spaces, as in the previous post, is based on the fact that the Yoneda lemma lets us think of sheaves as describing all the “probes” of a generalized space (actually an algebra in this case). The “probes” are the objects of a certain category, and are called “superpoints“.

This category is just \mathbf{Spt} = \mathbf{Gr}^{op}, the opposite of the category of Grassman algebras (i.e. exterior algebras) – that is, polynomial algebras in noncommuting variables, like \Lambda(\theta_1,\dots,\theta_q). These objects naturally come with a \mathbb{Z}_2-grading, which are spanned, respectively, by the monomials with even and odd degree: \Lambda_q = latex \mathbf{SMan}$ (\Lambda_q)_0 \oplus (\Lambda_q)_1$

(\Lambda_q)_0 = span( 1, \theta_i \theta_j, \theta_{i_1}\dots\theta{i_4}, \dots )


(\Lambda_q)_1 = span( \theta_i, \theta_i \theta_j \theta_k, \theta_{i_1}\dots\theta_{i_5},\dots )

This is a \mathbb{Z}_2-grading since the even ones commute with anything, and the odd ones anti-commute with each other. So if f_i and f_j are homogeneous (live entirely in one grade or the other), then f_i f_j = (-1)^{deg(i)deg(j)} f_j f_i.

The \Lambda_q should be thought of as the (0|q)-dimensional supermanifold: it looks like a point, with a q-dimensional fermionic tangent space (the “infinitesimal fuzz” noted above) attached. The morphisms in \mathbf{Spt} from \Lambda_q to $llatex \Lambda_r$ are just the grade-preserving algebra homomorphisms from \Lambda_r to \Lambda_q. There are quite a few of these: these objects are not terminal objects like the actual point. But this makes them good probes. Thi gets to be a site with the trivial topology, so that all presheaves are sheaves.

Then, as usual, a presheaf M on this category is to be understood as giving, for each object A=\Lambda_q, the collection of maps from \Lambda_q to a space M. The case q=0 gives the set of points of M, and the various other algebras A give sets of “A-points”. This term is based on the analogy that a point of a topological space (or indeed element of a set) is just the same as a map from the terminal object 1, the one point space (or one element set). Then an “A-point” of a space X is just a map from another object A. If A is not terminal, this is close to the notion of a “subspace” (though a subspace, strictly, would be a monomorphism from A). These are maps from A in \mathbf{Spt} = \mathbf{Gr}^{op}, or as algebra maps, M_A consists of all the maps \mathcal{O}(M) \rightarrow A.

What’s more, since this is a functor, we have to have a system of maps between the M_A. For any algebra maps A \rightarrow A', we should get corresponding maps M_{A'} \rightarrow M_A. These are really algebra maps \Lambda_q \rightarrow \Lambda_{q'}, of which there are plenty, all determined by the images of the generators \theta_1, \dots, \theta_q.

Now, really, a sheaf on \mathbf{Spt} is actually just what we might call a “super-set”, with sets M_A for each A \in \mathbf{Spt}. To make super-manifolds, one wants to say they are “manifold-valued sheaves”. Since manifolds themselves don’t form a topos, one needs to be a bit careful about defining the extra structure which makes a set a manifold.

Thus, a supermanifold M is a manifold constructed in the topos Sh(\mathbf{Spt}). That is, M must also be equipped with a topology and a collection of charts defining the manifold structure. These are all construed internally using objects and morphisms in the category of sheaves, where charts are based on super-domains, namely those algebras which look like C^{\infty}(V) \otimes \Lambda_q, for V an open subset of \mathbb{R}^p.

The reduced manifold M_{red} which appears in Batchelor’s theorem is the manifold of ordinary points M_{\mathbb{R}}. That is, it is all the \mathbb{R}-points, where \mathbb{R} is playing the role of functions on the zero-dimensional domain with just one point. All the extra structure in an atlas of charts for all of M to make it a supermanifold amounts to putting the structure of ordinary manifolds on the M_A – but in compatible ways.

(Alternatively, we could have described \mathbf{SMan} as sheaves in Sh(\mathbf{SDom}), where \mathbf{SDom} is a site of “superdomains”, and put all the structure defining a manifold into \mathbf{SDom}. But working over super-points is preferable for the moment, since it makes it clear that manifolds and supermanifolds are just manifestations of the same basic definition, but realized in two different toposes.)

The fact that the manifold structure on the M_A must be put on them compatibly means there is a relatively nice way to picture all these spaces.

Values of the Functor of Points as Bundles

The main idea which I find helps to understand the functor of points is that, for every superpoint \mathbb{R}^{0|n} (i.e. for every Grassman algebra A=\Lambda_n), one gets a manifold M_A. (Note the convention that q is the odd dimension of M, and n is the odd dimension of the probe superpoint).

Just as every supermanifold is a bundle of superpoints, every manifold M_A is a perfectly conventional vector bundle over the conventional manifold M_{red} of ordinary points. So for each A, we get a bundle, M_A \rightarrow M_{red}.

Now this manifold, M_{red}, consists exactly of all the “points” of M – this tells us immediately that \mathbf{SMan} is not a category of concrete sheaves (in the sense I explained in the previous post). Put another way, it’s not a concrete category – that would mean that there is an underlying set functor, which gives a set for each object, and that morphisms are determined by what they do to underlying sets. Non-concrete categories are, by nature, trickier to understand.

However, the functor of points gives a way to turn the non-concrete M into a tower of concrete manifolds M_A, and the morphisms between various M amount to compatible towers of maps between the various M_A for each A. The fact that the compatibility is controlled by algebra maps \Lambda_q \rightarrow \Lambda_{q'} explains why this is the same as maps between these bundles of superpoints.

Specifically, then, we have

M_A = \{ \mathcal{O}(M) \rightarrow A \}

This splits into maps of the even parts, and of the odd parts, where the grassman algebra A = \Lambda_n has even and odd parts: A = A_0 \oplus A_1, as above. Similarly, \mathcal{O}(M) splits into odd and even parts, and since the functions on M_{red} are entirely even, this is:

( \mathcal{O}(M))_0 = C^{\infty}(M_{red}) \otimes ( \Lambda_q)_0


( \mathcal{O}(M))_1 = C^{\infty}(M_{red}) \otimes (\Lambda_q)_1)

Now, the duality of “hom” and tensor means that Hom(\mathcal{O}(M),A) \cong \mathcal{O}(M) \otimes A, and algebra maps preserve the grading. So we just have tensor products of these with the even and odd parts, respectively, of the probe superpoint. Since the even part A_0 includes the multiples of the constants, part of this just gives a copy of U itself. The remaining part of A_0 is nilpotent (since it’s made of even-degree polynomials in the nilpotent \theta_i, so what we end up with, looking at the bundle over an open neighborhood U \subset M_{red}, is:

U_A = U \times ( (\Lambda_q)_0 \otimes A^{nil}_0) \times ((\Lambda_q)_1 \otimes A_1)

The projection map U_A \rightarrow U is the obvious projection onto the first factor. These assemble into a bundle over M_{red}.

We should think of these bundles as “shifting up” the nilpotent part of M (which are invisible at the level of ordinary points in M_{red}) by the algebra A. Writing them this way makes it clear that this is functorial in the superpoints A = \Lambda_n: given choices n and n', and any morphism between the corresponding A and A', it’s easy to see how we get maps between these bundles.

Now, maps between supermanifolds are the same thing as natural transformations between the functors of points. These include maps of the base manifolds, along with maps between the total spaces of all these bundles. More, this tower of maps must commute with all those bundle maps coming from algebra maps A \rightarrow A'. (In particular, since A = \Lambda_0, the ordinary point, is one of these, they have to commute with the projection to M_{red}.) These conditions may be quite restrictive, but it leaves us with, at least, a quite concrete image of what maps of supermanifolds

Super-Poincaré Group

One of the main settings where super-geometry appears is in so-called “supersymmetric” field theories, which is a concept that makes sense when fields live on supermanifolds. Supersymmetry, and symmetries associated to super-Lie groups, is exactly the kind of thing that John has worked on. A super-Lie group, of course, is a supermanifold that has the structure of a group (i.e. it’s a Lie group in the topos of presheaves over the site of super-points – so the discussion above means it can be thought of as a big tower of Lie groups, all bundles over a Lie group G_{red}).

In fact, John has mostly worked with super-Lie algebras (and the connection between these and division algebras, though that’s another story). These are \mathbb{Z}_2-graded algebras with a Lie bracket whose commutation properties are the graded version of those for an ordinary Lie algebra. But part of the value of the framework above is that we can simply borrow results from Lie theory for manifolds, import it into the new topos PSh(\mathbf{Spt}), and know at once that super-Lie algebras integrate up to super-Lie groups in just the same way that happens in the old topos (of sets).

Supersymmetry refers to a particular example, namely the “super-Poincaré group”. Just as the Poincaré group is the symmetry group of Minkowski space, a 4-manifold with a certain metric on it, the super-Poincaré group has the same relation to a certain supermanifold. (There are actually a few different versions, depending on the odd dimension.) The algebra is generated by infinitesimal translations and boosts, plus some “translations” in fermionic directions, which generate the odd part of the algebra.

Now, symmetry in a quantum theory means that this algebra (or, on integration, the corresponding group) acts on the Hilbert space \mathcal{H} of possible states of the theory: that is, the space of states is actually a representation of this algebra. In fact, to make sense of this, we need a super-Hilbert space (i.e. a graded one). The even generators of the algebra then produce grade-preserving self-maps of \mathcal{H}, and the odd generators produce grade-reversing ones. (This fact that there are symmetries which flip the “bosonic” and “fermionic” parts of the total \mathcal{H} is why supersymmetric theories have “superpartners” for each particle, with the opposite parity, since particles are labelled by irreducible representations of the Poincaré group and the gauge group).

To date, so far as I know, there’s no conclusive empirical evidence that real quantum field theories actually exhibit supersymmetry, such as detecting actual super-partners for known particles. Even if not, however, it still has some use as a way of developing toy models of quite complicated theories which are more tractable than one might expect, precisely because they have lots of symmetry. It’s somewhat like how it’s much easier to study computationally difficult theories like gravity by assuming, for instance, spherical symmetry as an extra assumption. In any case, from a mathematician’s point of view, this sort of symmetry is just a particularly simple case of symmetries for theories which live on noncommutative backgrounds, which is quite an interesting topic in its own right. As usual, physics generates lots of math which remains both true and interesting whether or not it applies in the way it was originally suggested.

In any case, what the functor-of-points viewpoint suggests is that ordinary and super- symmetries are just two special cases of “symmetries of a field theory” in two different toposes. Understanding these and other examples from this point of view seems to give a different understanding of what “symmetry”, one of the most fundamental yet slippery concepts in mathematics and science, actually means.

This semester, Susama Agarwala and I have been sharing a lecture series for graduate students. (A caveat: there are lecture notes there, by student request, but they’re rough notes, and contain some mistakes, omissions, and represent a very selective view of the subject.) Being a “topics” course, it consists of a few different sections, loosely related, which revolve around the theme of categorical tools which are useful for geometry (and topology).

What this has amounted to is: I gave a half-semester worth of courses on toposes, sheaves, and the basics of derived categories. Susama is now giving the second half, which is about motives. This post will talk about the part of the course I gave. Though this was a whole series of lectures which introduced all these topics more or less carefully, I want to focus here on the part of the lecture which built up to a discussion of sheaves as spaces. Nothing here, or in the two posts to follow, is particularly new, but they do amount to a nice set of snapshots of some related ideas.

Coming up soon: John Huerta is currently visiting Hamburg, and on  July 8, he gave a guest-lecture which uses some of this machinery to talk about supermanifolds, which will be the subject of the next post in this series. In a later post, I’ll talk about Susama’s lectures about motives and how this relates to the discussion here (loosely).

Grothendieck Toposes

The first half of our course was about various aspects of Grothendieck toposes. In the first lecture, I talked about “Elementary” (or Lawvere-Tierney) toposes. One way to look at these is to say that they are categories \mathcal{E} which have all the properties of the category of Sets which make it useful for doing most of ordinary mathematics. Thus, a topos in this sense is a category with a bunch of properties – there are various equivalent definitions, but for example, toposes have all finite limits (in particular, products), and all colimits.

More particularly, they have “power objects”. That is, if A and B are objects of \mathcal{E}, then there is an object B^A, with an “evaluation map” B^A \times A \rightarrow B, which makes it possible to think of B^A as the object of “morphisms from A to B”.

The other main thing a topos has is a “subobject classifier”. Now, a subobject of A \in \mathcal{E} is an equivalence class of monomorphisms into A – think of sets, where this amounts to specifying the image, and the monomorphisms are the various inclusions which pick out the same subset as their image. A classifier for subobjects should be thought of as something like the two-element set is Sets, whose elements we can tall “true” and “false”. Then every subset of A corresponds to a characteristic function A \rightarrow \mathbf{2}. In general, a subobject classifies is an object \Omega together with a map from the terminal object, T : 1 \rightarrow \Omega, such that every inclusion of subobject is a pullback of T along a characteristic function.

Now, elementary toposes were invented chronologically later than Grothendieck toposes, which are a special class of example. These are categories of sheaves on (Grothendieck) sites. A site is a category \mathcal{T} together with a “topology” J, which is a rule which, for each U \in \mathcal{T}, picks out J(U), a set of collections of maps into U, called seives for U. They collections J(U) have to satisfy certain conditions, but the idea can be understood in terms of the basic example, \mathcal{T} = TOP(X). Given a topological space, TOP(X) is the category whose objects are the open sets U \subset X, and the morphisms are all the inclusions. Then  that each collection in J(U) is an open cover of U – that is, a bunch of inclusions of open sets, which together cover all of U in the usual sense.

(This is a little special to TOP(X), where every map is an inclusion – in a general site, the J(U) need to be closed under composition with any other morphism (like an ideal in a ring). So for instance, \mathcal{T} = Top, the category of topological spaces, the usual choice of J(U) consists of all collections of maps which are jointly surjective.)

The point is that a presheaf on \mathcal{T} is just a functor \mathcal{T}^{op} \rightarrow Sets. That is, it’s a way of assigning a set to each U \in \mathcal{T}. So, for instance, for either of the cases we just mentioned, one has B : \mathcal{T}^{op} \rightarrow Sets, which assigns to each open set U the set of all bounded functions on U, and to every inclusion the restriction map. Or, again, one has C : \mathcal{T}^{op} \rightarrow Sets, which assigns the set of all continuous functions.

These two examples illustrate the condition which distinguishes those presheaves S which are sheaves – namely, those which satisfy some “gluing” conditions. Thus, suppose we’re, given an open cover \{ f_i : U_i \rightarrow U \}, and a choice of one element x_i from each S(U_i), which form a “matching family” in the sense that they agree when restricted to any overlaps. Then the sheaf condition says that there’s a unique “amalgamation” of this family – that is, one element x \in S(U) which restricts to all the x_i under the maps S(f_i) : S(U) \rightarrow S(U_i).

Sheaves as Generalized Spaces

There are various ways of looking at sheaves, but for the purposes of the course on categorical methods in geometry, I decided to emphasize the point of view that they are a sort of generalized spaces.

The intuition here is that all the objects and morphisms in a site \mathcal{T} have corresponding objects and morphisms in Psh(\mathcal{T}). Namely, the objects appear as the representable presheaves, U \mapsto Hom(-,U), and the morphisms U \rightarrow V show up as the induced natural transformations between these functors. This map y : \mathcal{T} \rightarrow Psh(\mathcal{T}) is called the Yoneda embedding. If \mathcal{T} is at all well-behaved (as it is in all the examples we’re interested in here), these presheaves will always be sheaves: the image of y lands in Sh(\mathcal{T}).

In this case, the Yoneda embedding embeds \mathcal{T} as a sub-category of Sh(\mathcal{T}). What’s more, it’s a full subcategory: all the natural transformations between representable presheaves come from the morphisms of \mathcal{T}-objects in a unique way. So  Sh(\mathcal{T}) is, in this sense, a generalization of \mathcal{T} itself.

More precisely, it’s the Yoneda lemma which makes sense of all this. The idea is to start with the way ordinary \mathcal{T}-objects (from now on, just call them “spaces”) S become presheaves: they become functors which assign to each U the set of all maps into S. So the idea is to turn this around, and declare that even non-representable sheaves should have the same interpretation. The Yoneda Lemma makes this a sensible interpretation: it says that, for any presheaf F \in Psh(\mathcal{T}), and any U \in \mathcal{T}, the set F(U) is naturally isomorphic to Hom(y(U),F): that is, F(U) literally is the collection of morphisms from U (or rather, its image under the Yoneda embedding) and a “generalized space” F. (See also Tom Leinster’s nice discussion of the Yoneda Lemma if this isn’t familiar.) We describe U as a “probe” object: one probes the space F by mapping U into it in various ways. Knowing the results for all U \in \mathcal{T} tells you all about the “space” F. (Thus, for instance, one can get all the information about the homotopy type of a space if you know all the maps into it from spheres of all dimensions up to homotopy. So spheres are acting as “probes” to reveal things about the space.)

Furthermore, since Sh(\mathcal{T}) is a topos, it is often a nicer category than the one you start with. It has limits and colimits, for instance, which the original category might not have. For example, if the kind of spaces you want to generalize are manifolds, one doesn’t have colimits, such as the space you get by gluing together two lines at a point. The sheaf category does. Likewise, the sheaf category has exponentials, and manifolds don’t (at least not without the more involved definitions needed to allow infinite-dimensional manifolds).

These last remarks about manifolds suggest the motivation for the first example…

Diffeological Spaces

The lecture I gave about sheaves as spaces used this paper by John Baez and Alex Hoffnung about “smooth spaces” (they treat Souriau’s diffeological spaces, and the different but related Chen spaces in the same framework) to illustrate the point. They describe In that case, the objects of the sites are open (or, for Chen spaces, convex) subsets of \mathbb{R}^n, for all choices of n, the maps are the smooth maps in the usual sense (i.e. the sense to be generalized), and the covers are jointly surjective collections of maps.

Now, that example is a somewhat special situation: they talk about concrete sheaves, on concrete sites, and the resulting categories are only quasitoposes – a slightly weaker condition than being a topos, but one still gets a useful collection of spaces, which among other things include all manifolds. The “concreteness” condition – that \mathcal{T} has a terminal object to play the role of “the point”. Being a concrete sheaf then means that all the “generalized spaces” have an underlying set of points (namely, the set of maps from the point object), and that all morphisms between the spaces are completely determined by what they do to the underlying set of points. This means that the “spaces” really are just sets with some structure.

Now, if the site happens to be TOP(X), then we have a slightly intuition: the “generalized” spaces are something like generalized bundles over X, and the “probes” are now sections of such a bundle. A simple example would be an actual sheaf of functions: these are sections of a trivial bundle, since, say, \mathbb{C}-valued functions are sections of the bundle \pi: X \times \mathbb{C} \rightarrow X. Given a nontrivial bundle \pi : M \rightarrow X, there is a sheaf of sections – on each U, one gets F_M(U) to be all the one-sided inverses s : U \rightarrow M which are one-sided inverses of \pi. For a generic sheaf, we can imagine a sort of “generalized bundle” over X.


Another example of the fact that sheaves can be seen as spaces is the category of schemes: these are often described as topological spaces which are themselves equipped with a sheaf of rings. “Scheme” is to algebraic geometry what “manifold” is to differential geometry: a kind of space which looks locally like something classical and familiar. Schemes, in some neighborhood of each point, must resemble varieties – i.e. the locus of zeroes of some algebraic function on $\mathbb{k}^n$. For varieties, the rings attached to neighborhoods are rings of algebraic functions on this locus, which will be a quotient of the ring of polynomials.

But another way to think of schemes is as concrete sheaves on a site whose objects are varieties and whose morphisms are algebraic maps. This is dual to the other point of view, just as thinking of diffeological spaces as sheaves is dual to a viewpoint in which they’re seen as topological spaces equipped with a notion of “smooth function”.

(Some general discussion of this in a talk by Victor Piercey)


These two viewpoints (defining the structure of a space by a class of maps into it, or by a class of maps out of it) in principle give different definitions. To move between them, you really need everything to be concrete: the space has an underlying set, the set of probes is a collection of real set-functions. Likewise, for something like a scheme, you’d need the ring for any open set to be a ring of actual set-functions. In this case, one can move between the two descriptions of the space as long as there is a pre-existing concept of the right kind of function  on the “probe” spaces. Given a smooth space, say, one can define a sheaf of smooth functions on each open set by taking those whose composites with every probe are smooth. Conversely, given something like a scheme, where the structure sheaf is of function rings on each open subspace (i.e. the sheaf is representable), one can define the probes from varieties to be those which give algebraic functions when composed with every function in these rings. Neither of these will work in general: the two approaches define different categories of spaces (in the smooth context, see Andrew Stacey’s comparison of various categories of smooth spaces, defined either by specifying the smooth maps in, or out, or both). But for very concrete situations, they fit together neatly.

The concrete case is therefore nice for getting an intuition for what it means to think of sheaves as spaces. For sheaves which aren’t concrete, morphisms aren’t determined by what they do to the underlying points i.e. the forgetful “underlying set” functor isn’t faithful. Here, we might think of a “generalized space” which looks like two copies of the same topological space: the sheaf gives two different elements of F(U) for each map of underlying sets. We could think of such generalized space as built from sets equipped with extra “stuff” (say, a set consisting of pairs (x,i) \in X \times \{ blue , green \} – so it consists of a “blue” copy of X and a “green” copy of X, but the underlying set functor ignores the colouring.

Still, useful as they may be to get a first handle on this concept of sheaf as generalized space, one shouldn’t rely on these intuitions too much: if \mathcal{T} doesn’t even have a “point” object, there is no underlying set functor at all. Eventually, one simply has to get used to the idea of defining a space by the information revealed by probes.

In the next post, I’ll talk more about this in the context of John Huerta’s guest lecture, applying this idea to the category of supermanifolds, which can be seen as manifolds built internal to the topos of (pre)sheaves on a site whose objects are called “super-points”.

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.


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).


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.


Since I moved to Hamburg,   Alessandro Valentino and I have been organizing one series of seminar talks whose goal is to bring people (mostly graduate students, and some postdocs and others) up to speed on the tools used in Jacob Lurie’s big paper on the classification of TQFT and proof of the Cobordism Hypothesis.  This is part of the Forschungsseminar (“research seminar”) for the working groups of Christoph Schweigert, Ingo Runkel, and Christoph Wockel.  First, I gave one introducing myself and what I’ve done on Extended TQFT. In our main series We’ve had a series of four so far – two in which Alessandro outlined a sketch of what Lurie’s result is, and another two by Sebastian Novak and Marc Palm that started catching our audience up on the simplicial methods used in the theory of (\infty,n)-categories which it uses.  Coming up in the New Year, Nathan Bowler and I will be talking about first (\infty,1)-categories, and then (\infty,n)-categories.   I’ll do a few posts summarizing the talks around then.

Some people in the group have done some work on quantum field theories with defects, in relation to which, there’s this workshop coming up here in February!  The idea here is that one could have two regions of space where different field theories apply, which are connected along a boundary. We might imagine these are theories which are different approximations to what’s going on physically, with a different approximation useful in each region.  Whatever the intuition, the regions will be labelled by some category, and boundaries between regions are labelled by functors between categories.  Where different boundary walls meet, one can have natural transformations.  There’s a whole theory of how a 3D TQFT can be associated to modular tensor categories, in sort of the same sense that a 2D TQFT is associated to a Frobenius algebra. This whole program is intimately connected with the idea of “extending” a given TQFT, in the sense that it deals with theories that have inputs which are spaces (or, in the case of defects, sub-spaces of given ones) of many different dimensions.  Lurie’s paper describing the n-dimensional cobordism category, is very much related to the input to a theory like this.

Brno Visit

This time, I’d like to mention something which I began working on with Roger Picken in Lisbon, and talked about for the first time in Brno, Czech Republic, where I was invited to visit at Masaryk University.  I was in Brno for a week or so, and on Thursday, December 13, I gave this talk, called “Higher Gauge Theory and 2-Group Actions”.  But first, some pictures!

This fellow was near the hotel I stayed in:


Since this sculpture is both faceless and hard at work on nonspecific manual labour, I assume he’s a Communist-era artwork, but I don’t really know for sure.

The Christmas market was on in Náměstí Svobody (Freedom Square) in the centre of town.  This four-headed dragon caught my eye:


On the way back from Brno to Hamburg, I met up with my wife to spend a couple of days in Prague.  Here’s the Christmas market in the Old Town Square of Prague:


Anyway, it was a good visit to the Czech Republic.  Now, about the talk!

Moduli Spaces in Higher Gauge Theory

The motivation which I tried to emphasize is to define a specific, concrete situation in which to explore the concept of “2-Symmetry”.  The situation is supposed to be, if not a realistic physical theory, then at least one which has enough physics-like features to give a good proof of concept argument that such higher symmetries should be meaningful in nature.  The idea is that Higher Gauge theory is a field theory which can be understood as one in which the possible (classical) fields on a space/spacetime manifold consist of maps from that space into some target space X.  For the topological theory, they are actually just homotopy classes of maps.  This is somewhat related to Sigma models used in theoretical physics, and mathematically to Homotopy Quantum Field Theory, which considers these maps as geometric structure on a manifold.  An HQFT is a functor taking such structured manifolds and cobordisms into Hilbert spaces and linear maps.  In the paper Roger and I are working on, we don’t talk about this stage of the process: we’re just considering how higher-symmetry appears in the moduli spaces for fields of this kind, which we think of in terms of Higher Gauge Theory.

Ordinary topological gauge theory – the study of flat connections on G-bundles for some Lie group G, can be looked at this way.  The target space X = BG is the “classifying space” of the Lie group – homotopy classes of maps in Hom(M,BG) are the same as groupoid homomorphisms in Hom(\Pi_1(M),G).  Specifically, the pair of functors \Pi_1 and B relating groupoids and topological spaces are adjoints.  Now, this deals with the situation where X = BG is a homotopy 1-type, which is to say that it has a fundamental groupoid \Pi_1(X) = G, and no other interesting homotopy groups.  To deal with more general target spaces X, one should really deal with infinity-groupoids, which can capture the whole homotopy type of X – in particular, all its higher homotopy groups at once (and various relations between them).  What we’re talking about in this paper is exactly one step in that direction: we deal with 2-groupoids.

We can think of this in terms of maps into a target space X which is a 2-type, with nontrivial fundamental groupoid \Pi_1(X), but also interesting second homotopy group \pi_2(X) (and nothing higher).  These fit together to make a 2-groupoid \Pi_2(X), which is a 2-group if X is connected.  The idea is that X is the classifying space of some 2-group \mathcal{G}, which plays the role of the Lie group G in gauge theory.  It is the “gauge 2-group”.  Homotopy classes of maps into X = B \mathcal{G} correspond to flat connections in this 2-group.

For practical purposes, we use the fact that there are several equivalent ways of describing 2-groups.  Two very directly equivalent ways to define them are as group objects internal to \mathbf{Cat}, or as categories internal to \mathbf{Grp} – which have a group of objects and a group of morphisms, and group homomorphisms that define source, target, composition, and so on.  This second way is fairly close to the equivalent formulation as crossed modules (G,H,\rhd,\partial).  The definition is in the slides, but essentially the point is that G is the group of objects, and with the action G \rhd H, one gets the semidirect product G \ltimes H which is the group of morphisms.  The map \partial : H \rightarrow G makes it possible to speak of G and H acting on each other, and that these actions “look like conjugation” (the precise meaning of which is in the defining properties of the crossed module).

The reason for looking at the crossed-module formulation is that it then becomes fairly easy to understand the geometric nature of the fields we’re talking about.  In ordinary gauge theory, a connection can be described locally as a 1-form with values in Lie(G), the Lie algebra of G.  Integrating such forms along curves gives another way to describe the connection, in terms of a rule assigning to every curve a holonomy valued in G which describes how to transport something (generally, a fibre of a bundle) along the curve.  It’s somewhat nontrivial to say how this relates to the classic definition of a connection on a bundle, which can be described locally on “patches” of the manifold via 1-forms together with gluing functions where patches overlap.  The resulting categories are equivalent, though.

In higher gauge theory, we take a similar view. There is a local view of “connections on gerbes“, described by forms and gluing functions (the main difference in higher gauge theory is that the gluing functions related to higher cohomology).  But we will take the equivalent point of view where the connection is described by G-valued holonomies along paths, and H-valued holonomies over surfaces, for a crossed module (G,H,\rhd,\partial), which satisfy some flatness conditions.  These amount to 2-functors of 2-categories \Pi_2(M) \rightarrow \mathcal{G}.

The moduli space of all such 2-connections is only part of the story.  2-functors are related by natural transformations, which are in turn related by “modifications”.  In gauge theory, the natural transformations are called “gauge transformations”, and though the term doesn’t seem to be in common use, the obvious term for the next layer would be “gauge modifications”. It is possible to assemble a 2-groupoid Hom(\Pi_2(M),\mathcal{G}, whose space of objects is exactly the moduli space of 2-connections, and whose 1- and 2-morphisms are exactly these gauge transformations and modifications.  So the question is, what is the meaning of the extra information contained in the 2-groupoid which doesn’t appear in the moduli space itself?

Our claim is that this information expresses how the moduli space carries “higher symmetry”.

2-Group Actions and the Transformation Double Category

What would it mean to say that something exhibits “higher” symmetry? A rudimentary way to formalize the intuition of “symmetry” is to say that there is a group (of “symmetries”) which acts on some object. One could get more subtle, but this should be enough to begin with. We already noted that “higher” gauge theory uses 2-groups (and beyond into n-groups) in the place of ordinary groups.  So in this context, the natural way to interpret it is by saying that there is an action of a 2-group on something.

Just as there are several equivalent ways to define a 2-group, there are different ways to say what it means for it to have an action on something.  One definition of a 2-group is to say that it’s a 2-category with one object and all morphisms and 2-morphisms invertible.  This definition makes it clear that a 2-group has to act on an object of some 2-category \mathcal{C}. For our purposes, just as we normally think of group actions on sets, we will focus on 2-group actions on categories, so that \mathcal{C} = \mathbf{Cat} is the 2-category of interest. Then an action is just a map:

\Phi : \mathcal{G} \rightarrow \mathbf{Cat}

The unique object of \mathcal{G} – let’s call it \star, gets taken to some object \mathbf{C} = \Phi(\star) \in \mathbf{Cat}.  This object \mathbf{C} is the thing being “acted on” by \mathcal{G}.  The existence of the action implies that there are automorphisms \Phi(g) : \mathbf{C} \rightarrow \mathbf{C} for every morphism in \mathbf{G} (which correspond to the elements of the group G of the crossed module).  This would be enough to describe ordinary symmetry, but the higher symmetry is also expressed in the images of 2-morphisms \Phi( \eta : g \rightarrow g') = \Phi(\eta) : \Phi(g) \rightarrow \Phi(g'), which we might call 2-symmetries relating 1-symmetries.

What we want to do in our paper, which the talk summarizes, is to show how this sort of 2-group action gives rise to a 2-groupoid (actually, just a 2-category when the \mathbf{C} being acted on is a general category).  Then we claim that the 2-groupoid of connections can be seen as one that shows up in exactly this way.  (In the following, I have to give some credit to Dany Majard for talking this out and helping to find a better formalism.)

To make sense of this, we use the fact that there is a diagrammatic way to describe the transformation groupoid associated to the action of a group G on a set S.  The set of morphisms is built as a pullback of the action map, \rhd : (g,s) \mapsto g(s).


This means that morphisms are pairs (g,s), thought of as going from s to g(s).  The rule for composing these is another pullback.  The diagram which shows how it’s done appears in the slides.  The whole construction ends up giving a cubical diagram in \mathbf{Sets}, whose top and bottom faces are mere commuting diagrams, and whose four other faces are all pullback squares.

To construct a 2-category from a 2-group action is similar. For now we assume that the 2-group action is strict (rather than being given by \Phi a weak 2-functor).  In this case, it’s enough to think of our 2-group \mathcal{G} not as a 2-category, but as a group-object in \mathbf{Cat} – the same way that a 1-group, as well as being a category, can be seen as a group object in \mathbf{Set}.  The set of objects of this category is the group G of morphisms of the 2-category, and the morphisms make up the group G \ltimes H of 2-morphisms.  Being a group object is the same as having all the extra structure making up a 2-group.

To describe a strict action of such a \mathcal{G} on \mathbf{C}, we just reproduce in \mathbf{Cat} the diagram that defines an action in \mathbf{Sets}:


The fact that \rhd is an action just means this commutes. In principle, we could define a weak action, which would mean that this commutes up to isomorphism, but we won’t be looking at that here.

Constructing the same diagram which describes the structure of a transformation groupoid (p29 in the slides for the talk), we get a structure with a “category of objects” and a “category of morphisms”.  The construction in \mathbf{Set} gives us directly a set of morphisms, while S itself is the set of objects. Similarly, in \mathbf{Cat}, the category of objects is just \mathbf{C}, while the construction gives a category of morphisms.

The two together make a category internal to \mathbf{Cat}, which is to say a double category.  By analogy with S / \!\! / G, we call this double category \mathbf{C} / \!\! / \mathcal{G}.

We take \mathbf{C} as the category of objects, as the “horizontal category”, whose morphisms are the horizontal arrows of the double category. The category of morphisms of \mathbf{C} /\!\!/ \mathcal{G} shows up by letting its objects be the vertical arrows of the double category, and its morphisms be the squares.  These look like this:


The vertical arrows are given by pairs of objects (\gamma, x), and just like the transformation 1-groupoid, each corresponds to the fact that the action of \gamma takes x to \gamma \rhd x. Each square (morphism in the category of morphisms) is given by a pair ( (\gamma, \eta), f) of morphisms, one from \mathcal{G} (given by an element in G \rtimes H), and one from \mathbf{C}.

The horizontal arrow on the bottom of this square is:

(\partial \eta) \gamma \rhd f \circ \Phi(\gamma,\eta)_x = \Phi(\gamma,\eta)_y \circ \gamma \rhd f

The fact that these are equal is exactly the fact that \Phi(\gamma,\eta) is a natural transformation.

The double category \mathbf{C} /\!\!/ \mathcal{G} turns out to have a very natural example which occurs in higher gauge theory.

Higher Symmetry of the Moduli Space

The point of the talk is to show how the 2-groupoid of connections, previously described as Hom(\Pi_2(M),\mathcal{G}), can be seen as coming from a 2-group action on a category – the objects of this category being exactly the connections. In the slides above, for various reasons, we did this in a discretized setting – a manifold with a decomposition into cells. This is useful for writing things down explicitly, but not essential to the idea behind the 2-symmetry of the moduli space.

The point is that there is a category we call \mathbf{Conn}, whose objects are the connections: these assign G-holonomies to edges of our discretization (in general, to paths), and H-holonomies to 2D faces. (Without discretization, one would describe these in terms of Lie(G)-valued 1-forms and Lie(H)-valued 2-forms.)

The morphisms of \mathbf{Conn} are one type of “gauge transformation”: namely, those which assign H-holonomies to edges. (Or: Lie(H)-valued 1-forms). They affect the edge holonomies of a connection just like a 2-morphism in \mathcal{G}.  Face holonomies are affected by the H-value that comes from the boundary of the face.

What’s physically significant here is that both objects and morphisms of \mathbf{Conn} describe nonlocal geometric information.  They describe holonomies over edges and surfaces: not what happens at a point.  The “2-group of gauge transformations”, which we call \mathbf{Gauge}, on the other hand, is purely about local transformations.  If V is the vertex set of the discretized manifold, then \mathbf{Gauge} = \mathcal{G}^V: one copy of the gauge 2-group at each vertex.  (Keeping this finite dimensional and avoiding technical details was one main reason we chose to use a discretization.  In principle, one could also talk about the 2-group of \mathcal{G}-valued functions, whose objects and morphisms, thinking of it as a group object in \mathbf{Cat}, are functions valued in morphisms of \mathcal{G}.)

Now, the way \mathbf{Gauge} acts on \mathbf{Conn} is essentially by conjugation: edge holonomies are affected by pre- and post-multiplication by the values at the two vertices on the edge – whether objects or morphisms of \mathbf{Gauge}.  (Face holonomies are unaffected).  There are details about this in the slides, but the important thing is that this is a 2-group of purely local changes.  The objects of \mathbf{Gauge} are gauge transformations of this other type.  In a continuous setting, they would be described by G-valued functions.  The morphisms are gauge modifications, and could be described by H-valued functions.

The main conceptual point here is that we have really distinguished between two kinds of gauge transformation, which are the horizontal and vertical arrows of the double category \mathbf{Conn} /\!\!/ \mathbf{Gauge}.  This expresses the 2-symmetry by moving some gauge transformations into the category of connections, and others into the 2-group which acts on it.  But physically, we would like to say that both are “gauge transformations”.  So one way to do this is to “collapse” the double category to a bicategory: just formally allow horizontal and vertical arrows to compose, so that there is only one kind of arrow.  Squares become 2-cells.

So then if we collapse the double category expressing our 2-symmetry relation this way, the result is exactly equivalent to the functor category way of describing connections.  (The morphisms will all be invertible because \mathbf{Conn} is a groupoid and \mathbf{Gauge} is a 2-group).

I’m interested in this kind of geometrical example partly because it gives a good way to visualize something new happening here.  There appears to be some natural 2-symmetry on this space of fields, which is fairly easy to see geometrically, and distinguishes in a fundamental way between two types of gauge transformation.  This sort of phenomenon doesn’t occur in the world of \mathbf{Sets} – a set S has no morphisms, after all, so the transformation groupoid for a group action on it is much simpler.

In broad terms, this means that 2-symmetry has qualitatively new features that familiar old 1-symmetry doesn’t have.  Higher categorical versions – n-groups acting on n-groupoids, as might show up in more complicated HQFT – will certainly be even more complicated.  The 2-categorical version is just the first non-trivial situation where this happens, so it gives a nice starting point to understand what’s new in higher symmetry that we didn’t already know.

This entry is a by-special-request blog, which Derek Wise invited me to write for the blog associated with the International Loop Quantum Gravity Seminar, and it will appear over there as well.  The ILQGS is a long-running regular seminar which runs as a teleconference, with people joining in from various countries, on various topics which are more or less closely related to Loop Quantum Gravity and the interests of people who work on it.  The custom is that when someone gives a talk, someone else writes up a description of the talk for the ILQGS blog, and Derek invited me to write up a description of his talk.  The audio file of the talk itself is available in .aiff and .wav formats, and the slides are here.

The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a few papers (two shorter ones, “Spontaneously broken Lorentz symmetry for Hamiltonian gravity“, “Linking Covariant and Canonical General Relativity via Local Observers“, and a new, longer one called “Lifting General Relativity to Observer Space“).

The key idea behind this project is the notion of “observer space”, which is exactly what it sounds like: a space of all observers in a given universe.  This is easiest to picture when one has a spacetime – a manifold with a Lorentzian metric, (M,g) – to begin with.  Then an observer can be specified by choosing a particular point (x_0,x_1,x_2,x_3) = \mathbf{x} in spacetime, as well as a unit future-directed timelike vector v.  This vector is a tangent to the observer’s worldline at \mathbf{x}.  The observer space is therefore a bundle over M, the “future unit tangent bundle”.  However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying (M,g).

The result is a surprising, relatively new physical intuition is that “spacetime” is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same spacetime.  This is somewhat related to the relativity of locality, which I’ve blogged about previously.  Geometrically, it is similar to the fact that a slicing of spacetime into space and time is not unique, and not respected by the full symmetries of the theory of Relativity, even for flat spacetime (much less for the case of General Relativity).  Similarly, we will see a notion of “observer space”, which can sometimes be turned into a bundle over an objective spacetime M, but not in all cases.

So, how is this described mathematically?  In particular, what did I mean up there by saying that spacetime becomes observer-dependent?

Cartan Geometry

The answer uses Cartan geometry, which is a framework for differential geometry that is slightly broader than what is commonly used in physics.  Roughly, one can say “Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry”.  The more familiar direction of generalization here is the fact that, like Riemannian geometry, Cartan is concerned with manifolds which have local models in terms of simple, “flat” geometries, but which have curvature, and fail to be homogeneous.  First let’s remember how Klein geometry works.

Klein’s Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role.  It describes “homogeneous spaces”, which are geometries in which every point is indistinguishable from every other point.  This is expressed by the existence of a transitive action of some Lie group G of all symmetries on an underlying space.  Any given point x will be fixed by some symmetries, and not others, so one also has a subgroup H = Stab(x) \subset G.  This is the “stabilizer subgroup”, consisting of all symmetries which fix x.  That the space is homogeneous means that for any two points x,y, the subgroups Stab(x) and Stab(y) are conjugate (by a symmetry taking x to y).  Then the homogeneous space, or Klein geometry, associated to (G,H) is, up to isomorphism, just the same as the quotient space G/H of the obvious action of H on G.

The advantage of this program is that it has a great many examples, but the most relevant ones for now are:

  • n-dimensional Euclidean space. the Euclidean group ISO(n) = SO(n) \ltimes \mathbb{R}^n is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant.  It acts transitively on \mathbb{R}^n.  Any point will be fixed by the group of rotations centred at that point, which is a subgroup of ISO(n) isomorphic to SO(n).  Klein’s insight is to reverse this: we may define Euclidean space by R^n \cong ISO(n)/SO(n).
  • n-dimensional Minkowski space.  Similarly, we can define this space to be ISO(n-1,1)/SO(n-1,1).  The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same.
  • de Sitter space.  As a Klein geometry, this is the quotient SO(4,1)/SO(3,1).  That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point.  But the global symmetries of de Sitter space are different.  Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations so(4,1)/so(3,1) and iso(3,1)/so(3,1) are identical, seen as representations of SO(3,1).  It’s natural to identify them with the tangent space at a point.  de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in \mathbb{R}^5.  This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature.
  • anti-de Sitter space. This is similar, but now the quotient is SO(3,2)/SO(3,1) – in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative.

Now, what does it mean to say that a Cartan geometry has a local model?  Well, just as a Lorentzian or Riemannian manifold is “locally modelled” by Minkowski or Euclidean space, a Cartan geometry is locally modelled by some Klein geometry.  This is best described in terms of a connection on a principal G-bundle, and the associated G/H-bundle, over some manifold M.  The crucial bundle in a Riemannian or Lorenztian geometry is the frame bundle: the fibre over each point consists of all the ways to isometrically embed a standard Euclidean or Minkowski space into the tangent space.  A connection on this bundle specifies how this embedding should transform as one moves along a path.  It’s determined by a 1-form on M, valued in the Lie algebra of G.

Given a parametrized path, one can apply this form to the tangent vector at each point, and get a Lie algebra-valued answer.  Integrating along the path, we get a path in the Lie group G (which is independent of the parametrization).  This is called a “development” of the path, and by applying the G-values to the model space G/H, we see that the connection tells us how to move through a copy of G/H as we move along the path.  The image this suggests is of “rolling without slipping” – think of the case where the model space is a sphere.  The connection describes how the model space “rolls” over the surface of the manifold M.  Curvature of the connection measures the failure to commute of the processes of rolling in two different directions.  A connection with zero curvature describes a space which (locally at least) looks exactly like the model space: picture a sphere rolling against its mirror image.  Transporting the sphere-shaped fibre around any closed curve always brings it back to its starting position. Now, curvature is defined in terms of transports of these Klein-geometry fibres.  If curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model.

This idea, that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics.

Gravity and Cartan Geometry

MacDowell-Mansouri gravity can be understood as a theory in which General Relativity is modelled by a Cartan geometry.  Of course, a standard way of presenting GR is in terms of the geometry of a Lorentzian manifold.  In the Palatini formalism, the basic fields are a connection A and a vierbein (coframe field) called e, with dynamics encoded in the Palatini action, which is the integral over M of R[\omega] \wedge e \wedge e, where R is the curvature 2-form for \omega.

This can be derived from a Cartan geometry, whose model geometry is de Sitter space SO(4,1)/SO(3,1).   Then MacDowell-Mansouri gravity gets \omega and e by splitting the Lie algebra as so(4,1) = so(3,1) \oplus \mathbb{R^4}.  This “breaks the full symmetry” at each point.  Then one has a fairly natural action on the so(4,1)-connection:

\int_M tr(F_h \wedge \star F_h)

Here, F_h is the so(3,1) part of the curvature of the big connection.  The splitting of the connection means that F_h = R + e \wedge e, and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action).  So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space.

The cosmological constant in GR shows up because a “flat” connection for a Cartan geometry based on de Sitter space will look (if measured by Minkowski space) as if it has constant curvature which is exactly that of the model Klein geometry.  The way to think of this is to take the fibre bundle of homogeneous model spaces as a replacement for the tangent bundle to the manifold.  The fibre at each point describes the local appearance of spacetime.  If empty spacetime is flat, this local model is Minkowski space, ISO(3,1)/SO(3,1), and one can really speak of tangent “vectors”.  The tangent homogeneous space is not linear.  In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of Lie(G)-valued forms can be treated in the same way, regardless of which Klein geometry G/H occurs in the fibres.  In particular, General Relativity, formulated in terms of Cartan geometry, always says that, in the absence of matter, the geometry of space is flat, and the cosmological constant is included naturally by the choice of which Klein geometry is the local model of spacetime.

Observer Space

The idea in defining an observer space is to combine two symmetry reductions into one.  The reduction from SO(4,1) to SO(3,1) gives de Sitter space, SO(4,1)/SO(3,1) as a model Klein geometry, which reflects the “symmetry breaking” that happens when choosing one particular point in spacetime, or event.  Then, the reduction of SO(3,1) to SO(3) similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector).  These are the tangent vectors to the worldline of an observer at the chosen point, so SO(3,1)/SO(3) the model Klein geometry, is the space of such possible observers.  The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in SO(3,1) translate between observers.  So locally, choosing an observer amounts to a splitting of the model spacetime at the point into a product of space and time. If we combine both reductions at once, we get the 7-dimensional Klein geometry SO(4,1)/SO(3).  This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers”

A general observer space O, however, is just a Cartan geometry modelled on SO(4,1)/SO(3).  This is a 7-dimensional manifold, equipped with the structure of a Cartan geometry.  One class of examples are exactly the future unit tangent bundles to 4-dimensional Lorentzian spacetimes.  In these cases, observer space is naturally a contact manifold: that is, it’s an odd-dimensional manifold equipped with a 1-form \alpha, the contact form, which is such that the top-dimensional form \alpha \wedge d \alpha \wedge \dots \wedge d \alpha is nowhere zero.  This is the odd-dimensional analog of a symplectic manifold.  Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane.  In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold M.

Now, Minkowski space has a slicing into space and time – in fact, one for each observer, who defines the time direction, but the time coordinate does not transform in any meaningful way under the symmetries of the theory, and different observers will choose different ones.  In just the same way, the homogeneous model of observer space can naturally be written as a bundle SO(4,1)/SO(3) \rightarrow SO(4,1)/SO(3,1).  But a general observer space O may or may not be a bundle over an ordinary spacetime manifold, O \rightarrow M.  Every Cartan geometry M gives rise to an observer space O as the bundle of future-directed timelike vectors, but not every Cartan geometry O is of this form, in any natural way. Indeed, without a further condition, we can’t even reconstruct observer space as such a bundle in an open neighborhood of a given observer.

This may be intuitively surprising: it gives a perfectly concrete geometric model in which “spacetime” is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between “space” and “time” in General Relativity. It may be impossible, that is, to determine objectively whether two observers are located at the same base event or not. This is a kind of “Relativity of Locality” which is geometrically much like the by-now more familiar Relativity of Simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree.  The coincident observers according to a given observer are those reached by a good class of geodesics in O moving only in directions that observer sees as boosts.

When one can reconstruct O \rightarrow M, two observers will agree whether or not they are coincident.  This extra condition which makes this possible is an integrability constraint on the action of the Lie algebra H (in our main example, H = SO(3,1)) on the observer space O.  In this case, the fibres of the bundle are the orbits of this action, and we have the familiar world of Relativity, where simultaneity may be relative, but locality is absolute.

Lifting Gravity to Observer Space

Apart from describing this model of relative spacetime, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) GR locally near each point in such an observer space.  The goal is to make a link between covariant and canonical quantization of gravity.  Covariant quantization treats the geometry of spacetime all at once, by means of a Lagrangian action functional.  This is mathematically appealing, since it respects the symmetry of General Relativity, namely its diffeomorphism-invariance.  On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one gets a Hilbert space by quantizing the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution.  This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer.  The point of observer space is that we consider all these choices at once.  Describing GR in O is both covariant, and based on (local) choices of time direction.

This is easiest to describe in the case of a bundle O \rightarrow M.  Then a “field of observers” to be a section of the bundle: a choice, at each base event in M, of an observer based at that event.  A field of observers may or may not correspond to a particular decomposition of spacetime into space evolving in time, but locally, at each point in O, it always looks like one.  The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer.  In this case, a Cartan connection on observer space is described by to a Lie(SO(4,1))-valued form.  This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation.  The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of SO(4,1) to SO(3,1), which symmetry breaking entails choosing a point); and second, between space and time (i.e. the reduction of SO(3,1) to SO(3), which symmetry breaking entails choosing a time direction).

This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift GR to a theory on an observer space O.  This amount to describing fields on O and an action functional, so that the splitting of the fields gives back the usual fields of GR on spacetime, and the action gives back the usual action.  This part of the project is still under development, but this lifting has been described.  In the case when there is no “objective” spacetime, the result includes some surprising new fields which it’s not clear how to deal with, but when there is an objective spacetime, the resulting theory looks just like GR.