### Why Higher Geometric Quantization

The largest single presentation was a pair of talks on “The Motivation for Higher Geometric Quantum Field Theory” by Urs Schreiber, running to about two and a half hours, based on these notes. This was probably the clearest introduction I’ve seen so far to the motivation for the program he’s been developing for several years. Broadly, the idea is to develop a higher-categorical analog of geometric quantization (GQ for short).

One guiding idea behind this is that we should really be interested in quantization over (higher) stacks, rather than merely spaces. This leads inexorably to a higher-categorical version of GQ itself. The starting point, though, is that the defining features of stacks capture two crucial principles from physics: the gauge principle, and locality. The gauge principle means that we need to keep track not just of connections, but gauge transformations, which form respectively the objects and morphisms of a groupoid. “Locality” means that these groupoids of configurations of a physical field on spacetime is determined by its local configuration on regions as small as you like (together with information about how to glue together the data on small regions into larger regions).

Some particularly simple cases can be described globally: a scalar field gives the space of all scalar functions, namely maps into $\mathbb{C}$; sigma models generalise this to the space of maps $\Sigma \rightarrow M$ for some other target space. These are determined by their values pointwise, so of course are local.

More generally, physicists think of a field theory as given by a fibre bundle $V \rightarrow \Sigma$ (the previous examples being described by trivial bundles $\pi : M \times \Sigma \rightarrow \Sigma$), where the fields are sections of the bundle. Lagrangian physics is then described by a form on the jet bundle of $V$, i.e. the bundle whose fibre over $p \in \Sigma$ consists of the space describing the possible first $k$ derivatives of a section over that point.

More generally, a field theory gives a procedure $F$ for taking some space with structure – say a (pseudo-)Riemannian manifold $\Sigma$ – and produce a moduli space $X = F(\Sigma)$ of fields. The Sigma models happen to be representable functors: $F(\Sigma) = Maps(\Sigma,M)$ for some $M$, the representing object. A prestack is just any functor taking $\Sigma$ to a moduli space of fields. A stack is one which has a “descent condition”, which amounts to the condition of locality: knowing values on small neighbourhoods and how to glue them together determines values on larger neighborhoods.

The Yoneda lemma says that, for reasonable notions of “space”, the category $\mathbf{Spc}$ from which we picked target spaces $M$ embeds into the category of stacks over $\mathbf{Spc}$ (Riemannian manifolds, for instance) and that the embedding is faithful – so we should just think of this as a generalization of space. However, it’s a generalization we need, because gauge theories determine non-representable stacks. What’s more, the “space” of sections of one of these fibred stacks is also a stack, and this is what plays the role of the moduli space for gauge theory! For higher gauge theories, we will need higher stacks.

All of the above is the classical situation: the next issue is how to quantize such a theory. It involves a generalization of Geometric Quantization (GQ for short). Now a physicist who actually uses GQ will find this perspective weird, but it flows from just the same logic as the usual method.

In ordinary GQ, you have some classical system described by a phase space, a manifold $X$ equipped with a pre-symplectic 2-form $\omega \in \Omega^2(X)$. Intuitively, $\omega$ describes how the space, locally, can be split into conjugate variables. In the phase space for a particle in $n$-space, these “position” and “momentum” variables, and $\omega = \sum_x dx^i \wedge dp^i$; many other systems have analogous conjugate variables. But what really matters is the form $\omega$ itself, or rather its cohomology class.

Then one wants to build a Hilbert space describing the quantum analog of the system, but in fact, you need a little more than $(X,\omega)$ to do this. The Hilbert space is a space of sections of some bundle whose sections look like copies of the complex numbers, called the “prequantum line bundle“. It needs to be equipped with a connection, whose curvature is a 2-form in the class of $\omega$: in general, . (If $\omega$ is not symplectic, i.e. is degenerate, this implies there’s some symmetry on $X$, in which case the line bundle had better be equivariant so that physically equivalent situations correspond to the same state). The easy case is the trivial bundle, so that we get a space of functions, like $L^2(X)$ (for some measure compatible with $\omega$). In general, though, this function-space picture only makes sense locally in $X$: this is why the choice of prequantum line bundle is important to the interpretation of the quantized theory.

Since the crucial geometric thing here is a bundle over the moduli space, when the space is a stack, and in the context of higher gauge theory, it’s natural to seek analogous constructions using higher bundles. This would involve, instead of a (pre-)symplectic 2-form $\omega$, an $(n+1)$-form called a (pre-)$n$-plectic form (for an introductory look at this, see Chris Rogers’ paper on the case $n=2$ over manifolds). This will give a higher analog of the Hilbert space.

Now, maps between Hilbert spaces in QG come from Lagrangian correspondences – these might be maps of moduli spaces, but in general they consist of a “space of trajectories” equipped with maps into a space of incoming and outgoing configurations. This is a span of pre-symplectic spaces (equipped with pre-quantum line bundles) that satisfies some nice geometric conditions which make it possible to push a section of said line bundle through the correspondence. Since each prequantum line bundle can be seen as maps out of the configuration space into a classifying space (for $U(1)$, or in general an $n$-group of phases), we get a square. The action functional is a cell that fills this square (see the end of 2.1.3 in Urs’ notes). This is a diagrammatic way to describe the usual GQ construction: the advantage is that it can then be repeated in the more general setting without much change.

This much is about as far as Urs got in his talk, but the notes go further, talking about how to extend this to infinity-stacks, and how the Dold-Kan correspondence tells us nicer descriptions of what we get when linearizing – since quantization puts us into an Abelian category.

I enjoyed these talks, although they were long and Urs came out looking pretty exhausted, because while I’ve seen several others on this program, this was the first time I’ve seen it discussed from the beginning, with a lot of motivation. This was presumably because we had a physically-minded part of the audience, whereas I’ve mostly seen these for mathematicians, and usually they come in somewhere in the middle and being more time-limited miss out some of the details and the motivation. The end result made it quite a natural development. Overall, very helpful!

The main thing happening in my end of the world is that it’s relocated from Europe back to North America. I’m taking up a teaching postdoc position in the Mathematics and Computer Science department at Mount Allison University starting this month. However, amidst all the preparations and moving, I was also recently in Edinburgh, Scotland for a workshop on Higher Gauge Theory and Higher Quantization, where I gave a talk called 2-Group Symmetries on Moduli Spaces in Higher Gauge Theory. That’s what I’d like to write about this time.

Edinburgh is a beautiful city, though since the workshop was held at Heriot-Watt University, whose campus is outside the city itself, I only got to see it on the Saturday after the workshop ended. However, John Huerta and I spent a while walking around, and as it turned out, climbing a lot: first the Scott Monument, from which I took this photo down Princes Street:

And then up a rather large hill called Arthur’s Seat, in Holyrood Park next to the Scottish Parliament.

The workshop itself had an interesting mix of participants. Urs Schreiber gave the most mathematically sophisticated talk, and mine was also quite category-theory-minded. But there were also some fairly physics-minded talks that are interesting to me as well because they show the source of these ideas. In this first post, I’ll begin with my own, and continue with David Roberts’ talk on constructing an explicit string bundle. …

### 2-Group Symmetries of Moduli Spaces

My own talk, based on work with Roger Picken, boils down to a couple of observations about the notion of symmetry, and applies them to a discrete model in higher gauge theory. It’s the kind of model you might use if you wanted to do lattice gauge theory for a BF theory, or some other higher gauge theory. But the discretization is just a convenience to avoid having to deal with infinite dimensional spaces and other issues that don’t really bear on the central point.

Part of that point was described in a previous post: it has to do with finding a higher analog for the relationship between two views of symmetry: one is “global” (I found the physics-inclined part of the audience preferred “rigid”), to do with a group action on the entire space; the other is “local”, having to do with treating the points of the space as objects of a groupoid who show how points are related to each other. (Think of trying to describe the orbit structure of just the part of a group action that relates points in a little neighborhood on a manifold, say.)

In particular, we’re interested in the symmetries of the moduli space of connections (or, depending on the context, flat connections) on a space, so the symmetries are gauge transformations. Now, here already some of the physically-inclined audience objected that these symmetries should just be eliminated by taking the quotient space of the group action. This is based on the slogan that “only gauge-invariant quantities matter”. But this slogan has some caveats: in only applies to closed manifolds, for one. When there are boundaries, it isn’t true, and to describe the boundary we need something which acts as a representation of the symmetries. Urs Schreiber pointed out a well-known example: the Chern-Simons action, a functional on a certain space of connections, is not gauge-invariant. Indeed, the boundary terms that show up due to this not-invariance explain why there is a Wess-Zumino-Witt theory associated with the boundaries when the bulk is described by Chern-Simons.

Now, I’ve described a lot of the idea of this talk in the previous post linked above, but what’s new has to do with how this applies to moduli spaces that appear in higher gauge theory based on a 2-group $\mathcal{G}$. The points in these space are connections on a manifold $M$. In particular, since a 2-group is a group object in categories, the transformation groupoid (which captures global symmetries of the moduli space) will be a double category. It turns out there is another way of seeing this double category by local descriptions of the gauge transformations.

In particular, general gauge transformations in HGT are combinations of two special types, described geometrically by $G$-valued functions, or $Lie(H)$-valued 1-forms, where $G$ is the group of objects of $\mathcal{G}$, and $H$ is the group of morphisms based at $1_G$. If we think of connections as functors from the fundamental 2-groupoid $\Pi_2(M)$ into $\mathcal{G}$, these correspond to pseudonatural transformations between these functors. The main point is that there are also two special types of these, called “strict”, and “costrict”. The strict ones are just natural transformations, where the naturality square commutes strictly. The costrict ones, also called ICONs (for “identity component oplax natural transformations” – see the paper by Steve Lack linked from the nlab page above for an explanation of “costrictness”). They assign the identity morphism to each object, but the naturality square commutes only up to a specified 2-cell. Any pseudonatural transformation factors into a strict and costrict part.

The point is that taking these two types of transformation to be the horizontal and vertical morphisms of a double category, we get something that very naturally arises by the action of a big 2-group of symmetries on a category. We also find something which doesn’t happen in ordinary gauge theory: that only the strict gauge transformations arise from this global symmetry. The costrict ones must already be the morphisms in the category being acted on. This category plays the role of the moduli space in the normal 1-group situation. So moving to 2-groups reveals that in general we should distinguish between global/rigid symmetries of the moduli space, which are strict gauge transformations, and costrict ones, which do not arise from the global 2-group action and should be thought of as intrinsic to the moduli space.

### String Bundles

David Roberts gave a rather interesting talk called “Constructing Explicit String Bundles”. There are some notes for this talk here. The point is simply to give an explicit construction of a particular 2-group bundle. There is a lot of general abstract theory about 2-bundles around, and a fair amount of work that manipulates physically-motivated descriptions of things that can presumably be modelled with 2-bundles. There has been less work on giving a mathematically rigorous description of specific, concrete 2-bundles.

This one is of interest because it’s based on the String 2-group. Details are behind that link, but roughly the classifying space of $String(G)$ (a homotopy 2-type) is fibred over the classifying space for $G$ (a 1-type). The exact map is determined by taking a pullback along a certain characteristic class (which is a map out of $BG$). Saying “the” string 2-group is a bit of a misnomer, by the way, since such a 2-group exists for every simply connected compact Lie group $G$. The group that’s involved here is a $String(n)$, the string 2-group associated to $Spin(n)$, the universal cover of the rotation group $SO(n)$. This is the one that determines whether a given manifold can support a “string structure”. A string structure on $M$, therefore, is a lift of a spin structure, which determines whether one can have a spin bundle over $M$, hence consistently talk about a spin connection which gives parallel transport for spinor fields on $M$. The string structure determines if one can consistently talk about a string-bundle over $M$, and hence a 2-group connection giving parallel transport for strings.

In this particular example, the idea was to find, explicitly, a string bundle over Minkowski space – or its conformal compactification. In point of fact, this particular one is for $latek String(5)$, and is over 6-dimensional Minkowski space, whose compactification is $M = S^5 \times S^1$. This particular $M$ is convenient because it’s possible to show abstractly that it has exactly one nontrivial class of string bundles, so exhibiting one gives a complete classification. The details of the construction are in the notes linked above. The technical details rely on the fact that we can coordinatize $M$ nicely using the projective quaternionic plane, but conceptually it relies on the fact that $S^5 \cong SU(3)/SU(2)$, and because of how the lifting works, this is also $String(SU(3))/String(SU(2))$. This quotient means there’s a string bundle $String(SU(3)) \rightarrow S^5$ whose fibre is $String(SU(2))$.

While this is only one string bundle, and not a particularly general situation, it’s nice to see that there’s a nice elegant presentation which gives such a bundle explicitly (by constructing cocycles valued in the crossed module associated to the string 2-group, which give its transition functions).

(Here endeth Part I of this discussion of the workshop in Edinburgh. Part II will talk about Urs Schreiber’s very nice introduction to Higher Geometric Quantization)

(This ends the first part of this update – the next will describe the physics-oriented talks, and the third will describe Urs Schreiber’s series on higher geometric quantization)

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

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

### Global and Local Symmetry

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

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

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

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

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

### Transformation Groupoid

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

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

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

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

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

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

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

### Categorify Everything

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

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

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

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

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

### 2-Group Actions

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

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

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

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

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

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

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

### The Transformation Double Category of a 2-Group Action

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

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

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

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

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

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

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

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

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

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

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

### Speculative Remarks

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

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

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

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

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

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

### Structured Sheaves and Internal Constructions in Toposes

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

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

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

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

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

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

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

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

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

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”.

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