To continue from the previous post

Twisted Differential Cohomology

Ulrich Bunke gave a talk introducing differential cohomology theories, and Thomas Nikolaus gave one about a twisted version of such theories (unfortunately, perhaps in the wrong order). The idea here is that cohomology can give a classification of field theories, and if we don’t want the theories to be purely topological, we would need to refine this. A cohomology theory is a (contravariant) functorial way of assigning to any space X, which we take to be a manifold, a \mathbb{Z}-graded group: that is, a tower of groups of “cocycles”, one group for each n, with some coboundary maps linking them. (In some cases, the groups are also rings) For example, the group of differential forms, graded by degree.

Cohomology theories satisfy some axioms – for example, the Mayer-Vietoris sequence has to apply whenever you cut a manifold into parts. Differential cohomology relaxes one axiom, the requirement that cohomology be a homotopy invariant of X. Given a differential cohomology theory, one can impose equivalence relations on the differential cocycles to get a theory that does satisfy this axiom – so we say the finer theory is a “differential refinement” of the coarser. So, in particular, ordinary cohomology theories are classified by spectra (this is related to the Brown representability theorem), whereas the differential ones are represented by sheaves of spectra – where the constant sheaves represent the cohomology theories which happen to be homotopy invariants.

The “twisting” part of this story can be applied to either an ordinary cohomology theory, or a differential refinement of one (though this needs similarly refined “twisting” data). The idea is that, if R is a cohomology theory, it can be “twisted” over X by a map \tau: X \rightarrow Pic_R into the “Picard group” of R. This is the group of invertible R-modules (where an R-module means a module for the cohomology ring assigned to X) – essentially, tensoring with these modules is what defines the “twisting” of a cohomology element.

An example of all this is twisted differential K-theory. Here the groups are of isomorphism classes of certain vector bundles over X, and the twisting is particularly simple (the Picard group in the topological case is just \mathbb{Z}_2). The main result is that, while topological twists are classified by appropriate gerbes on X (for K-theory, U(1)-gerbes), the differential ones are classified by gerbes with connection.

Fusion Categories

Scott Morrison gave a talk about Classifying Fusion Categories, the point of which was just to collect together a bunch of results constructing particular examples. The talk opens with a quote by Rutherford: “All science is either physics or stamp collecting” – that is, either about systematizing data and finding simple principles which explain it, or about collecting lots of data. This talk was unabashed stamp-collecting, on the grounds that we just don’t have a lot of data to systematically understand yet – and for that very reason I won’t try to summarize all the results, but the slides are well worth a look-over. The point is that fusion categories are very useful in constructing TQFT’s, and there are several different constructions that begin “given a fusion category \mathcal{C}“… and yet there aren’t all that many examples, and very few large ones, known.

Scott also makes the analogy that fusion categories are “noncommutative finite groups” – which is a little confusing, since not all finite groups are commutative anyway – but the idea is that the symmetric fusion categories are exactly the representation categories of finite groups. So general fusion categories are a non-symmetric generalization of such groups. Since classifying finite groups turned out to be difficult, and involve a laundry-list of sporadic groups, it shouldn’t be too surprising that understanding fusion categories (which, for the symmetric case, include the representation categories of all these examples) should be correspondingly tricky. Since, as he points out, we don’t have very many non-symmetric examples beyond rank 12 (analogous to knowing only finite groups with at most 12 elements), it’s likely that we don’t have a very good understanding of these categories in general yet.

There were a couple of talks – one during the workshop by Sonia Natale, and one the previous week by Sebastian Burciu, whom I also had the chance to talk with that week – about “Equivariantization” of fusion categories, and some fairly detailed descriptions of what results. The two of them have a paper on this which gives more details, which I won’t summarize – but I will say a bit about the construction.

An “equivariantization” of a category C acted on by a group G is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set.  The category C^G has objects which consist of an object x \in C which is fixed by the action of G, together with an isomorphism \mu_g : x \rightarrow x for each g \in G, satisfying a bunch of unsurprising conditions like being compatible with the group operation. The morphisms are maps in C between the objects, which form commuting squares for each g \in G. Their paper, and the talks, described how this works when C is a fusion category – namely, C^G is also a fusion category, and one can work out its fusion rules (i.e. monoidal structure). In some cases, it’s a “group theoretical” fusion category (it looks like Rep(H) for some group H) – or a weakened version of such a thing (it’s Morita equivalent to ).

A nice special case of this is if the group action happens to be trivial, so that every object of C is a fixed point. In this case, C^G is just the category of objects of C equipped with a G-action, and the intertwining maps between these. For example, if C = Vect, then C^G = Rep(G) (in particular, a “group-theoretical fusion category”). What’s more, this construction is functorial in G itself: given a subgroup H \subset G, we get an adjoint pair of functors between C^G and C^H, which in our special case are just the induced-representation and restricted-representation functors for that subgroup inclusion. That is, we have a Mackey functor here. These generalize, however, to any fusion category C, and to nontrivial actions of G on C. The point of their paper, then, is to give a good characterization of the categories that come out of these constructions.

Quantizing with Higher Categories

The last talk I’d like to describe was by Urs Schreiber, called Linear Homotopy Type Theory for Quantization. Urs has been giving evolving talks on this topic for some time, and it’s quite a big subject (see the long version of the notes above if there’s any doubt). However, I always try to get a handle on these talks, because it seems to be describing the most general framework that fits the general approach I use in my own work. This particular one borrows a lot from the language of logic (the “linear” in the title alludes to linear logic).

Basically, Urs’ motivation is to describe a good mathematical setting in which to construct field theories using ingredients familiar to the physics approach to “field theory”, namely… fields. (See the description of Kevin Walker’s talk.) Also, Lagrangian functionals – that is, the notion of a physical action. Constructing TQFT from modular tensor categories, for instance, is great, but the fields and the action seem to be hiding in this picture. There are many conceptual problems with field theories – like the mathematical meaning of path integrals, for instance. Part of the approach here is to find a good setting in which to locate the moduli spaces of fields (and the spaces in which path integrals are done). Then, one has to come up with a notion of quantization that makes sense in that context.

The first claim is that the category of such spaces should form a differentially cohesive infinity-topos which we’ll call \mathbb{H}. The “infinity” part means we allow morphisms between field configurations of all orders (2-morphisms, 3-morphisms, etc.). The “topos” part means that all sorts of reasonable constructions can be done – for example, pullbacks. The “differentially cohesive” part captures the sort of structure that ensures we can really treat these as spaces of the suitable kind: “cohesive” means that we have a notion of connected components around (it’s implemented by having a bunch of adjoint functors between spaces and points). The “differential” part is meant to allow for the sort of structures discussed above under “differential cohomology” – really, that we can capture geometric structure, as in gauge theories, and not just topological structure.

In this case, we take \mathbb{H} to have objects which are spectral-valued infinity-stacks on manifolds. This may be unfamiliar, but the main point is that it’s a kind of generalization of a space. Now, the sort of situation where quantization makes sense is: we have a space (i.e. \mathbb{H}-object) of field configurations to start, then a space of paths (this is WHERE “path-integrals” are defined), and a space of field configurations in the final system where we observe the result. There are maps from the space of paths to identify starting and ending points. That is, we have a span:

A \leftarrow X \rightarrow B

Now, in fact, these may all lie over some manifold, such as B^n(U(1)), the classifying space for U(1) (n-1)-gerbes. That is, we don’t just have these “spaces”, but these spaces equipped with one of those pieces of cohomological twisting data discussed up above. That enters the quantization like an action (it’s WHAT you integrate in a path integral).

Aside: To continue the parallel, quantization is playing the role of a cohomology theory, and the action is the twist. I really need to come back and complete an old post about motives, because there’s a close analogy here. If quantization is a cohomology theory, it should come by factoring through a universal one. In the world of motives, where “space” now means something like “scheme”, the target of this universal cohomology theory is a mild variation on just the category of spans I just alluded to. Then all others come from some functor out of it.

Then the issue is what quantization looks like on this sort of scenario. The Atiyah-Singer viewpoint on TQFT isn’t completely lost here: quantization should be a functor into some monoidal category. This target needs properties which allow it to capture the basic “quantum” phenomena of superposition (i.e. some additivity property), and interference (some actual linearity over \mathbb{C}). The target category Urs talked about was the category of E_{\infty}-rings. The point is that these are just algebras that live in the world of spectra, which is where our spaces already lived. The appropriate target will depend on exactly what \mathbb{H} is.

But what Urs did do was give a characterization of what the target category should be LIKE for a certain construction to work. It’s a “pull-push” construction: see the link way above on Mackey functors – restriction and induction of representations are an example . It’s what he calls a “(2-monoidal, Beck-Chevalley) Linear Homotopy-Type Theory”. Essentially, this is a list of conditions which ensure that, for the two morphisms in the span above, we have a “pull” operation for some and left and right adjoints to it (which need to be related in a nice way – the jargon here is that we must be in a Wirthmuller context), satisfying some nice relations, and that everything is functorial.

The intuition is that if we have some way of getting a “linear gadget” out of one of our configuration spaces of fields (analogous to constructing a space of functions when we do canonical quantization over, let’s say, a symplectic manifold), then we should be able to lift it (the “pull” operation) to the space of paths. Then the “push” part of the operation is where the “path integral” part comes in: many paths might contribute to the value of a function (or functor, or whatever it may be) at the end-point of those paths, because there are many ways to get from A to B, and all of them contribute in a linear way.

So, if this all seems rather abstract, that’s because the point of it is to characterize very generally what has to be available for the ideas that appear in physics notions of path-integral quantization to make sense. Many of the particulars – spectra, E_{\infty}-rings, infinity-stacks, and so on – which showed up in the example are in a sense just placeholders for anything with the right formal properties. So at the same time as it moves into seemingly very abstract terrain, this approach is also supposed to get out of the toy-model realm of TQFT, and really address the trouble in rigorously defining what’s meant by some of the standard practice of physics in field theory by analyzing the logical structure of what this practice is really saying. If it turns out to involve some unexpected math – well, given the underlying issues, it would have been more surprising if it didn’t.

It’s not clear to me how far along this road this program gets us, as far as dealing with questions an actual physicist would like to ask (for the most part, if the standard practice works as an algorithm to produce results, physicists seldom need to ask what it means in rigorous math language), but it does seem like an interesting question.

So I spent a few weeks at the Erwin Schrodinger Institute in Vienna, doing a short residence as part of the program “Modern Trends in Topological Quantum Field Theory” leading up to a workshop this week. There were quite a few interesting talks – some on topics that I’ve written about elsewhere in this blog, so I’ll gloss over those. For example, Catherine Meusburger spoke about the project with Barrett and Schaumann to give a diagrammatic language for Gray categories with duals – I’ve written about John Barrett’s talks on this elsewhere. Similarly, I’ve written about Chris Schommer-Pries’ talks about fully-extended TQFT’s and the cobordism hypothesis for structured cobordisms . I’d like to just describe some of the other highlights that connect nicely to themes I find interesting. In Part 1 of this post, the more topological themes…

TQFTs with Boundary

On the first day, Kevin Walker gave a talk called “Premodular TQFTs” which was quite interesting. The key idea here is that a fairly big class of different constructions of 3D TQFT’s turn out to actually be aspects of one 4D TQFT, which comes about by a construction based on the 3D construction of Crane-Yetter-Kauffman.  The term “premodular” refers to the fact that 3D TQFT’s can be related to modular tensor categories. “Tensor” includes several concepts, like being abelian, having vector spaces of morphisms, a monoidal structure that gets along with these – typical examples being the categories of vector spaces, or of representations of some fixed group. “Modular” means that there is a braiding, and that a certain string diagram (which looks like two linked rings) built using the braiding can be represented as an invertible matrix. These will show up as a special case of the “premodular” theory.

The basic idea is to use an approach that is based on local fields (which respects the physics-land concept of what “field theory” means), avoids the path integral approach (which is hard to make rigorous), and can be shown to connect back to the Atyiah-Singer approach in which a TQFT is a kind of functor out of a cobordism category.

That is, given a manifold X we must be able to find the fields on X, called F(X). For example, F(X) could be the maps into a classifying space BG, for a gauge theory, or a category of diagrams on X with labels in some appropriate sort of category. Then one has some relations which say when given fields are the same. For each manifold Y, this defines a vector space of linear combinations of fields, modulo relations, called A(Y;c), where c \in F(\partial Y). The dual space of A(Y;c) is called Z(Y;c) – in keeping with the principle that quantum states are functionals that we can evaluate on “classical” fields.

Walker’s talk develops, from this starting point, a view that includes a whole range of theories – the Dijkgraaf-Witten model (fields are maps to BG); diagrams in a semisimple 1-category (“Euler characteristic theory”), in a pivotal 2-category (a Turaev-Viro model), or a premodular 3-category (a “Crane-Yetter model”), among others. In particular, some familiar theories appear as living on 3D boundaries to a 4D manifold, where such a  premodular theory is defined. The talk goes on to describe a kind of “theory with defects”, where two different theories live on different parts of a manifold (this is a common theme to a number of the talks), and in particular it describes a bimodule which gives a Morita equivalence between two sorts of theory – one based on graphs labelled in representations of a group G, and the other based on G-connections. The bimodule is, effectively, a kind of “Fourier transform” which relates dimension-k structures on one side to codimension-k structures on the other: a line labelled by a G-representation on one side gets acted upon by G-holonomies for a hypersurface on the other side.

On a related note Alessandro Valentino gave a talk called “Boundary Conditions for 3d TQFT and module categories” This related to a couple of papers with Jurgen Fuchs and Christoph Schweigert. The basic idea starts with the fact that one can build (3,2,1)-dimensional TQFT’s from modular tensor categories \mathcal{C}, getting a Reshitikhin-Turaev type theory which assigns \mathcal{C} to the circle. The modular tensor structure tells you what gets assigned to higher-dimensional cobordisms. (This is a higher-categorical analog of the fact that a (2,1)-dimensional TQFT is determined by a Frobenius algebra). Then the motivating question is: how can we extend this theory all the way down to a point (i.e. have it assign something to a point, so that \mathcal{C} is somehow composed of naturally occurring morphisms).

So the question is: if we know what \mathcal{C} is, what does that tell us about the “colours” that could be assigned to a boundary. There’s a fairly elegant way to take on this question by looking at what’s assigned to Wilson lines, the observables that matter in defining RT-type theories, when the line where we’re observing gets pushed onto the boundary. (See around p14 of the first paper linked above). The colours on lines inside the manifold could be objects of \mathcal{C}, and fusing them illustrates the monoidal structure of \mathcal{C}. Then the question is what kind of category can be attached to a boundary and be consistent with this.This should be functorial with respect to fusing two lines (i.e. doing this before or after projecting to the boundary should be the same).

They don’t completely characterize the situation, but they give some reasonable arguments which suggest that the result is that the boundary category, a braided monoidal category, ought to be the Drinfel’d centre of something. This is actually a stronger constraint for categories than groups (any commutative group is the centre of something – namely itself – but this isn’t true for monoidal categories).

2-Knots

Joost Slingerland gave a talk called “Local Representations of the Loop Braid Group”, which was quite nice. The Loop Braid Group was introduced by the late Xiao-Song Lin (whom I had the pleasure to know at UCR) as an interesting generalization of the braid group B_n. B_n is the “motion group” of isomorphism classes of motions of n particles in a plane: in such a motion, we let the particles move around arbitrarily, before ending up occupying the same points occupied initially. (In the “pure braid group”, each individual point must end up where it started – in the braid group, they can swap places). Up to diffeomorphism, this keeps track of how they move around each other – not just how they exchange places, but which one crosses in front of which, etc. The loop braid group does the same for loops embedded in 3D space. Now, if the loops always stay far away from each other, one possibility is that a motion amounts to a permutation in which the loops switch places: two paths through 3D space (or 4D spacetime) can always be untangled. On the other hand, loops can pass THROUGH each other, as seen at the beginning of this video:

This is analogous to two points braiding in 2D space (i.e. strands twisting around each other in 3D spacetime), although in fact these “slide moves” form a group which is different from just the pure braid group – but PB_n fits inside them. In particular, the slide moves satisfy some of the same relations as the braid group – the Yang-Baxter equations.

The final thing that can happen is that loops might move, “flip over”, and return to their original position with reversed orientation. So the loop braid group can be broken down as LB_n = Slide_n \rtimes (\mathbb{Z}_2)^n \rtimes S_n. Every loop braid could be “closed up” to a 4D knotted surface, though not every knotted surface would be of this form. For one thing, our loops have a trivial embedding in 3D space here – to get every possible knotted surface, we’d need to have knots and links sliding around, braiding through each other, merging and splitting, etc. Knotted surfaces are much more complex than knotted circles, just as the topology of embedded circles is more complex than that of embedded points.

The talk described some work on the “local representations” of LB_n: representations on spaces where each loop is attached some k-dimensional vector space V (this is the “local dimension”), so that the motions of n loops gets represented on V^{\otimes n} (a tensor product of n copies of V). This is already rather complex, but is much easier than looking for arbitrary representations of LB_n on any old vector space (“nonlocal” representations, if you like). Now, in particular, for local dimension 2, this boils down to some simple matrices which can be worked out – the slide moves are either represented by some permutation matrices, or some tensor products of rotation matrices, or a few other cases which can all be classified.

Toward the end, Dror Bar-Natan also gave a talk that touched on knotted surfaces, called “A Partial Reduction of BF Theory to Combinatorics“. The mention of BF theory – a kind of higher gauge theory that can be described locally in terms of a 1-form and a 2-form on a manifold – is basically to set up some discussion of knotted surfaces (the combinatorics it reduces to). The point is that, like many field theories, BF theory amplitudes can be calculated using a sum over certain Feynman diagrams – but these ones are diagrams that lie partly in certain knotted surfaces. (See the rather remarkable handout in the link above for lots of pictures). This is sort of analogous to how some gauge theories in 3D boil down to knot invariants – for knots that live on the boundary of a region cut out of the 3-manifold. This is similar, for a knotted surface in a 4-manifold.

The “combinatorics” boils down to showing some diagram presentations of these knotted surfaces – particularly, a special type called a “ribbon knot”, which is a certain kind of knotted sphere. The combinatorics show that these special knotted surfaces all correspond to ordinary knotted circles in 3D (in the handout, you’ll see the Gauss diagram for a knot – a picture which shows which points along a line cross over or under each other in a presentation of the knot – used to construct a corresponding ribbon knot). But do check out the handout for some pictures which show several different ways of presenting 2-knots.

(…To be continued in Part 2…)

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:

actionsquare

(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:

squarepic

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}))

and

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

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

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

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

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

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

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

Dold-Puppe Correspondence

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

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

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

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

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

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

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

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

Chain Homotopies and Quasi-Isomorphisms

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

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

given by the quotients

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

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

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

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

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

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

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

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

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

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

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

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

Derived Categories

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

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

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

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

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

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

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

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

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

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

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

Supermanifolds

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

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

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

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

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

In this, V is as above, and the notation

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

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

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

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

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

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

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

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

Supermanifolds as Manifold-Valued Sheaves

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

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

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

and

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

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

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

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

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

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

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

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

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

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

Values of the Functor of Points as Bundles

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

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

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

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

Specifically, then, we have

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

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

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

and

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

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

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

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

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

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

Super-Poincaré Group

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

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

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

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

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

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

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

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

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

Grothendieck Toposes

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

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

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

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

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

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

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

Sheaves as Generalized Spaces

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

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

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

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

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

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

Diffeological Spaces

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

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

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

Schemes

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

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

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

Generalities

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

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

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

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

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

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

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

Cobordism Hypothesis

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

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

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

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

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

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

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

(\infty,n)-Categories

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

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

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

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

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

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

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

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

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

Cobordism Category as (\infty,n)-Category

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

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

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

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

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

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

Dualizability

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

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

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

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

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

and a coevaluation:

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

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

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

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

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

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

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

Classifying TQFT

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

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

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