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 ; sigma models generalise this to the space of maps 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 (the previous examples being described by trivial bundles ), where the fields are sections of the bundle. Lagrangian physics is then described by a form on the jet bundle of , i.e. the bundle whose fibre over consists of the space describing the possible first derivatives of a section over that point.

More generally, a field theory gives a procedure for taking some space with structure – say a (pseudo-)Riemannian manifold – and produce a moduli space of fields. The Sigma models happen to be *representable functors*: for some , the representing object. A prestack is just any functor taking 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 from which we picked target spaces embeds into the category of stacks over (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 equipped with a pre-symplectic 2-form . Intuitively, describes how the space, locally, can be split into conjugate variables. In the phase space for a particle in -space, these “position” and “momentum” variables, and ; many other systems have analogous conjugate variables. But what really matters is the form 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 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 : in general, . (If is not symplectic, i.e. is degenerate, this implies there’s some symmetry on , 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 (for some measure compatible with ). In general, though, this function-space picture only makes sense locally in : 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 , an -form called a (pre-)-plectic form (for an introductory look at this, see Chris Rogers’ paper on the case 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 , or in general an -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!

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Thomas Strobl’s talk, “New Methods in Gauge Theory” (based on a whole series of papers linked to from the conference webpage), started with a discussion of of generalizing Sigma Models. Strobl’s talk was a bit high-level physics for me to do it justice, but I came away with the impression of a fairly large program that has several points of contact with more mathematical notions I’ll discuss later.

In particular, Sigma models are physical theories in which a field configuration on spacetime is a map into some target manifold, or rather , since we need a metric to integrate and find differentials. Given this, we can define the crucial physics ingredient, an action functional

where the are the differentials of the map into .

In string theory, is the world-sheet of a string and is ordinary spacetime. This generalizes the simpler example of a moving particle, where is just its worldline. In that case, minimizing the action functional above says that the particle moves along geodesics.

The big generalization introduced is termed a “Dirac Sigma Model” or DSM (the paper that introduces them is this one).

In building up to these DSM, a different generalization notes that if there is a group action that describes “rigid” symmetries of the theory (for Minkowski space we might pick the Poincare group, or perhaps the Lorentz group if we want to fix an origin point), then the action functional on the space is invariant in the direction of any of the symmetries. One can use this to reduce , by “gauging out” the symmetries to get a quotient , and get a corresponding to integrate over .

To generalize this, note that there’s an action groupoid associated with , and replace this with some other (Poisson) groupoid instead. That is, one thinks of the real target for a gauge theory not as , but the action groupoid , and then just considers replacing this with some generic groupoid that doesn’t necessarily arise from a group of rigid symmetries on some underlying . (In this regard, see the second post in this series, about Urs Schreiber’s talk, and stacks as classifying spaces for gauge theories).

The point here seems to be that one wants to get a nice generalization of this situation – in particular, to be able to go backward from to , to deal with the possibility that the quotient may be geometrically badly-behaved. Or rather, given , to find some of which it is a reduction, but which is better behaved. That means needing to be able to treat a Sigma model with symmetry information attached.

There’s also an infinitesimal version of this: locally, invariance means the Lie derivative of the action in the direction of any of the generators of the Lie algebra of – so called Killing vectors – is zero. So this equation can generalize to a case where there are vectors where the Lie derivative is zero – a so-called “generalized Killing equation”. They may not generate isometries, but can be treated similarly. What they do give, if you integrate these vectors, is a foliation of . The space of leaves is the quotient mentioned above.

The most generic situation Thomas discussed is when one has a Dirac structure on – this is a certain kind of subbundle of the tangent-plus-cotangent bundle over .

Another couple of physics-y talks related higher gauge theory to some particular physics models, namely and supersymmetric field theories.

The first, by Martin Wolf, was called “Self-Dual Higher Gauge Theory”, and was rooted in generalizing some ideas about twistor geometry – here are some lecture notes by the same author, about how twistor geometry relates to ordinary gauge theory.

The idea of twistor geometry is somewhat analogous to the idea of a Fourier transform, which is ultimately that the same space of fields can be described in two different ways. The Fourier transform goes from looking at functions on a position space, to functions on a frequency space, by way of an integral transform. The Penrose-Ward transform, analogously, transforms a space of fields on Minkowski spacetime, satisfying one set of equations, to a set of fields on “twistor space”, satisfying a different set of equations. The theories represented by those fields are then equivalent (as long as the PW transform is an isomorphism).

The PW transform is described by a “correspondence”, or “double fibration” of spaces – what I would term a “span”, such that both maps are fibrations:

The general story of such correspondences is that one has some geometric data on , which we call – a set of functions, differential forms, vector bundles, cohomology classes, etc. They are pulled back to , and then “pushed forward” to by a direct image functor. In many cases, this is given by an integral along each fibre of the fibration , so we have an integral transform. The image of we call , and it consists of data satisfying, typically, some PDE’s.In the case of the PW transform, is complex projective 3-space and is the set of holomorphic principal bundles for some group ; is (complexified) Minkowski space and the fields are principal -bundles with connection. The PDE they satisfy is , where is the curvature of the bundle and is the Hodge dual). This means cohomology on twistor space (which classifies the bundles) is related self-dual fields on spacetime. One can also find that a point in corresponds to a projective line in , while a point in corresponds to a null plane in . (The space ).

Then the issue to to generalize this to higher gauge theory: rather than principal -bundles for a group, one is talking about a 2-group with connection. Wolf’s talk explained how there is a Penrose-Ward transform between a certain class of higher gauge theories (on the one hand) and an supersymmetric field theory (on the other hand). Specifically, taking , and to be (a subspace of) 6D projective space , there is a similar correspondence between certain holomorphic 2-bundles on and solutions to some self-dual field equations on (which can be seen as constraints on the curvature 3-form for a principal 2-bundle: the self-duality condition is why this only makes sense in 6 dimensions).

This picture generalizes to supermanifolds, where there are fermionic as well as bosonic fields. These turn out to correspond to a certain 6-dimensional supersymmetric field theory.

Then Sam Palmer gave a talk in which he described a somewhat similar picture for an supersymmetric theory. However, unlike the theory, this one gives, not a higher gauge theory, but something that superficially looks similar, but in fact is quite different. It ends up being a theory of a number of fields – form valued in three linked vector spaces

equipped with a bunch of maps that give the whole setup some structure. There is a collection of seven fields in groups (“multiplets”, in physics jargon) valued in each of these spaces. They satisfy a large number of identities. It somewhat resembles the higher gauge theory that corresponds to the case, so this situation gets called a “-gauge model”.

There are some special cases of such a setup, including Courant-Dorfman algebras and Lie 2-algebras. The talk gave quite a few examples of solutions to the equations that fall out. The overall conclusion is that, while there are some similarities between -gauge models and the way Higher Gauge Theory appears at the level of algebra-valued forms and the equations they must satisfy, there are some significant differences. I won’t try to summarize this in more depth, because (a) I didn’t follow the nitty-gritty technical details very well, and (b) it turns out to be not HGT, but some new theory which is less well understood at summary-level.

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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. …

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 . The points in these space are connections on a manifold . 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 -valued functions, or -valued 1-forms, where is the group of objects of , and is the group of morphisms based at . If we think of connections as functors from the fundamental 2-groupoid into , 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.

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 (a homotopy 2-type) is fibred over the classifying space for (a 1-type). The exact map is determined by taking a pullback along a certain characteristic class (which is a map out of ). 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 . The group that’s involved here is a , the string 2-group associated to , the universal cover of the rotation group . This is the one that determines whether a given manifold can support a “string structure”. A string structure on , therefore, is a lift of a spin structure, which determines whether one can have a spin bundle over , hence consistently talk about a spin connection which gives parallel transport for spinor fields on . The string structure determines if one can consistently talk about a string-bundle over , 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 . This particular 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 nicely using the projective quaternionic plane, but conceptually it relies on the fact that , and because of how the lifting works, this is also . This quotient means there’s a string bundle whose fibre is .

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)

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**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 , which we take to be a manifold, a -graded group: that is, a tower of groups of “cocycles”, one group for each , 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 . 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 is a cohomology theory, it can be “twisted” over by a map into the “Picard group” of . This is the group of invertible -modules (where an -module means a module for the cohomology ring assigned to ) – 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 , and the twisting is particularly simple (the Picard group in the topological case is just ). The main result is that, while topological twists are classified by appropriate gerbes on (for K-theory, -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 “… 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

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 acted on by a group is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set. The category has objects which consist of an object which is fixed by the action of , together with an isomorphism for each , satisfying a bunch of unsurprising conditions like being compatible with the group operation. The morphisms are maps in between the objects, which form commuting squares for each . Their paper, and the talks, described how this works when is a fusion category – namely, 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 for some group ) – 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 is a fixed point. In this case, is just the category of objects of equipped with a -action, and the intertwining maps between these. For example, if , then (in particular, a “group-theoretical fusion category”). What’s more, this construction is functorial in itself: given a subgroup , we get an adjoint pair of functors between and , 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 , and to nontrivial actions of on . 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 . 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 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. -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:

Now, in fact, these may all lie over some manifold, such as , the classifying space for -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 ). The target category Urs talked about was the category of -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 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, -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.

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**TQFTs with Boundary**

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

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

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

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

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

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

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

**2-Knots**

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

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

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

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

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

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

(…To be continued in Part 2…)

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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.

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 in some category 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 . More generally, we can talk about an action of a group on an object , which is a map .

“Local symmetry” is different, and it makes most sense in a context where the object is a set – or at least, where it makes sense to talk about elements of , so that 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 is faithful: you can tell morphisms in (in particular, automorphisms of ) apart by looking at what they do to the underlying set. The intuition is that the morphisms of are exactly set maps which preserve the structure which forgets about – or, conversely, that the structure on objects of is exactly that which is forgotten by . Certainly, knowing only this information determines up to equivalence. In any case, suppose we have an object like this: then knowing about the symmetries of amounts to knowing about a certain group action, namely the action of , on the underlying set .

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 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 , but a relationship under some specific symmetry between one element of and another. If there is an isomorphism between and , then and 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 : it may be that only part of has, for example, a reflection symmetry, but the symmetry doesn’t extend globally.

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 acting on a set . The result is called for short. Its morphisms consist of pairs such that is a morphism taking to its image under the action of . The “local” symmetry view of 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 .

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

(Here, is the multiplication in , and this is the familiar associativity-type axiom for a group action: acting by a product of two elements in 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 either is a set, or has an underlying set by way of some faithful forgetful functor: that is, every morphism in corresponds to a unique set map from the elements of to itself. We needed this to describe the groupoid , whose objects are exactly the elements of . The diagram above suggests a different way to think about this. The action diagram lives in the category : we are thinking of as a set together with some structure maps. and the morphism must be in the same category, , for this characterization to make sense.

So in fact, what matters is that the category lived in was *closed*: that is, it is enriched in itself, so that for any objects , there is an object , the *internal hom*. In this case, it’s which appears in the diagram. Such an internal hom is supposed to be a dual to ‘s monoidal product (which happens to be the Cartesian product ): this is exactly what lets us talk about .

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

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 , 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 -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 , and follow the same construction above. So our first ingredient is a 2-group . As with groups, we can think of a 2-group either as a 2-category with just one object , or as a 1-category with some structure – a group object in , which we’ll call 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 . Its objects are the morphisms of the 2-group, and the composition becomes the monoidal product .)

(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.)

So, again, there are two ways to talk about an action of a 2-group on some category . One is to define an action as a 2-functor . The object being acted on, , is the unique object – so that the 2-functor amounts to a monoidal functor from the categorical group into . Notice that here we’re taking advantage of the fact that is closed, so that the hom-“sets” are actually categories, and the automorphisms of – invertible functors from 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 , which corresponds to the 2-functor , and satisfies an action axiom like the square above, with 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 directly gives three maps. First, functors for each 2-group morphism – each of which consists of a function between objects of , together with a function between morphisms of . Second, natural transformations for 2-morphisms in the 2-group – each of which consists of a function from objects to morphisms of .

On the other hand, 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 is just given by . The map taking pairs of morphisms to morphisms of is less intuitively obvious. Since I already claimed and are equivalent, it should be no surprise that we ought to be able to reconstruct the other two parts of from it as special cases. These are morphism-maps for the functors, (which give or ), and the natural transformation maps (which give or ). In fact, there are only two sensible ways to combine these four bits of information, and the fact that is natural means precisely that they’re the same, so:

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 on the objects of , and of the morphisms of on objects of . They fit together nicely because objects can be identified with their identity morphisms: furthermore, being a functor gives an action of -objects on -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 ?

The answer is that a category (such as a groupoid) internal to is a *double category.* The compact way to describe it is as a “category in “, 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 , and the morphism-category is , and the target functor is just the action map . (The other structure maps that make this into a category in 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 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).

So by construction, the horizontal category of these squares is just the object-category . For the same reason, the squares and vertical morphisms, make up the category .

On the other hand, the vertical category has the same objects as , 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 -objects on the set of -objects, . Meanwhile, the horizontal morphisms and squares make up the transformation groupoid . 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 by adding 2-morphisms to it, corresponding to morphisms of : roughly, it makes the morphisms of into the objects of a new transformation groupoid, based on the action of the group of automorphisms of the identity in (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 and essentially determine the entire 2-group: the constraint only relates to the structure of .

The constraint for the vertical bicategory is different in flavour because it depends more on the action . Here we are extending a transformation groupoid, . But, for some actions, many morphisms in might just not show up at all. For 1-morphisms , the only 2-morphisms which can appear are those taking to some which has the same effect on as . So, for example, this will look very different if 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.

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 -gerbes and the corresponding -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.

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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.

The definition of a (pre)sheaf as a functor valued in is the basic one, but there are parallel notions for presheaves valued in categories other than – 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 ” is equivalent to a category of “abelian-group-valued sheaves on “, denoted . (As usual, I’ll omit the Grothendieck topology 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 , , or .

To begin with, we look at the category , which amounts to the same as the category of abelian group objects in . This inherits several properties from 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 , the direct sum of sheaves of abelian group just gives , and all the properties hold locally at each .

So, sheaves of abelian groups can be seen as abelian groups in a topos of sheaves . 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 , or sheaves of simplicial sets, . (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 are functors – that is, -valued presheaves on , the simplex category. This has nonnegative integers as its objects, and the morphisms from to are the order-preserving functions from to . If , we get “simplicial sets”, where is the “set of -dimensional simplices”. The various morphisms in 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 can also be seen as sheaves on valued in the category of simplicial sets, i.e. objects of . 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:

and

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

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 is a sequence of Abelian groups with boundary maps (or just for short), like so:

with the property that . Morphisms between these are sequences of morphisms between the terms of the complexes where each which commute with all the boundary maps. These all assemble into a category of complexes . We also have and , the (full) subcategories of complexes where all the negative (respectively, positive) terms are trivial.

One can generalize this to replace 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 , the last two examples, simplicial objects and complexes, are secretly the same thing.

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

**Theorem**: In any abelian category , the category of simplicial objects is equivalent to the category of positive chain complexes .

The better-known name “Dold-Kan Theorem” refers to the case where . If is a category of -valued sheaves, the Dold-Puppe correspondence amounts to using Dold-Kan at each .

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, . On objects, it gives the intersection of all the kernels of the face maps:

The boundary map from this is then just . 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 from complexes to simplicial objects in is defined so as to be adjoint to . Indeed, and together form an adjoint equivalence of the categories.

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 the space of smooth differential -forms on some smooth manifold , 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:

given by the quotients

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 , since there are natural transformations between these functors

which just come from the restrictions of the to the kernel . (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 gives a corresponding .

Now, the original motivation of cohomology for a space, like the de Rahm cohomology of a manifold , is to measure something about the topology of . If is trivial (say, a contractible space), then its cohomology groups are all trivial. In the general setting, we say that is *acyclic* if all the . 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 is a *quasi-isomorphism* if 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 is a quasi-isomorphism, it is a homotopy equivalence.

That is, it has a homotopy inverse , which means there is a homotopy .

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 is a collection of maps which shift degrees, , such that . That is, the coboundary of is the difference between and . (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…

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 is like internalizing the homotopy theory of spaces in a category . 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 . In the background, keep in mind the typical example where , and even where for some reasonably nice space , if it helps to picture things. Then the derived category of is built up in a few steps:

- Take the category of complexes. (This stands in for “spaces in ” 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.) - Take morphisms only up to homotopy equivalence. That is, define the equivalence relation with whenever there is a homotopy with . Then is the quotient by this relation.
- Localize at quasi-isomorphisms. That is, formally throw in inverses for all quasi-isomorphisms , to turn them into actual isomorphisms. The result is .

(Since we have direct sums of complexes (componentwise), it’s also possible to think of the last step as defining , where is the category of acyclic complexes – the ones whose cohomology complexes are zero.)

Explicitly, the morphisms of can be thought of as “zig-zags” in ,

where all the left-pointing arrows are quasi-isomorphisms. (The left-pointing arrows are standing in for their new inverses in , 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 , and hence about if 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.

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

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

Roughly, the idea is that supermanifolds are spaces like manifolds, but with some non-commuting coordinates. Supermanifolds are therefore in some sense “noncommutative spaces”. Noncommutative algebraic or differential geometry start with various dualities to the effect that some category of spaces is equivalent to the opposite of a corresponding category of algebras – for instance, a manifold corresponds to the algebra . So a generalized category of “spaces” can be found by dropping the “commutative” requirement from that statement. The category of supermanifolds only weakens the condition slightly: the algebras are -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 the ring 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 for some open set . (For fixed dimension ).

For supermanifolds, the condition on the local rings says that, for fixed dimension , a -dimensional supermanifold has structure sheaf in which $they look like

In this, is as above, and the notation

refers to the exterior algebra, which we can think of as polynomials in the , with the wedge product, which satisfies . The idea is that one is supposed to think of this as the algebra of smooth functions on a space with ordinary dimensions, and “anti-commuting” dimensions with coordinates . The commuting variables, say , 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, ).

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 .

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 . That is, sections in the local rings (“functions on” open neighborhoods of ) always glue together to give a section in .

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 , whose fibres are the “super-points” corresponding to . The base space 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 is a super “point”, when is a whole vector space? Since the fermionic variables are anticommuting, no term can have more than one of each , so this is a finite-dimensional algebra. This is unlike , which suggests that the noncommutative directions are quite different. Any element of is nilpotent, so if we think of a Taylor series for some function – a power series in the – we see note that no term has a coefficient for greater than 1, or of degree higher than in all the – so imagines that only infinitesimal behaviour in these directions exists at all. Thus, a supermanifold is like an ordinary -dimensional manifold , built from the ordinary domains , 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 .

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

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 , the opposite of the category of Grassman algebras (i.e. exterior algebras) – that is, polynomial algebras in noncommuting variables, like . These objects naturally come with a -grading, which are spanned, respectively, by the monomials with even and odd degree: latex \mathbf{SMan}$ (\Lambda_q)_0 \oplus (\Lambda_q)_1$

and

This is a -grading since the even ones commute with anything, and the odd ones anti-commute with each other. So if and are homogeneous (live entirely in one grade or the other), then .

The should be thought of as the -dimensional supermanifold: it looks like a point, with a -dimensional fermionic tangent space (the “infinitesimal fuzz” noted above) attached. The morphisms in from to $llatex \Lambda_r$ are just the grade-preserving algebra homomorphisms from to . 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 on this category is to be understood as giving, for each object , the collection of maps from to a space . The case gives the set of points of , and the various other algebras give sets of “-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 , the one point space (or one element set). Then an “-point” of a space is just a map from another object . If is not terminal, this is close to the notion of a “subspace” (though a subspace, strictly, would be a *monomorphism* from ). These are maps from in , or as algebra maps, consists of all the maps .

What’s more, since this is a functor, we have to have a system of maps between the . For any algebra maps , we should get corresponding maps . These are really algebra maps , of which there are plenty, all determined by the images of the generators .

Now, really, a sheaf on is actually just what we might call a “super-set”, with sets for each . 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 is a manifold constructed in the topos . That is, 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 , for an open subset of .

The reduced manifold which appears in Batchelor’s theorem is the manifold of ordinary points . That is, it is all the -points, where 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 to make it a supermanifold amounts to putting the structure of ordinary manifolds on the – but in compatible ways.

(Alternatively, we could have described as sheaves in , where is a site of “superdomains”, and put all the structure defining a manifold into . 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 must be put on them compatibly means there is a relatively nice way to picture all these spaces.

The main idea which I find helps to understand the functor of points is that, for every superpoint (i.e. for every Grassman algebra ), one gets a manifold . (Note the convention that is the odd dimension of , and is the odd dimension of the probe superpoint).

Just as every supermanifold is a bundle of superpoints, every manifold is a perfectly conventional vector bundle over the conventional manifold of* ordinary *points. So for each , we get a bundle, .

Now this manifold, , consists exactly of all the “points” of – this tells us immediately that 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 into a tower of concrete manifolds , and the morphisms between various amount to compatible towers of maps between the various for each . The fact that the compatibility is controlled by algebra maps explains why this is the same as maps between these bundles of superpoints.

Specifically, then, we have

This splits into maps of the even parts, and of the odd parts, where the grassman algebra has even and odd parts: , as above. Similarly, splits into odd and even parts, and since the functions on are entirely even, this is:

and

Now, the duality of “hom” and tensor means that , 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 includes the multiples of the constants, part of this just gives a copy of itself. The remaining part of is nilpotent (since it’s made of even-degree polynomials in the nilpotent , so what we end up with, looking at the bundle over an open neighborhood , is:

The projection map is the obvious projection onto the first factor. These assemble into a bundle over .

We should think of these bundles as “shifting up” the nilpotent part of (which are invisible at the level of ordinary points in ) by the algebra . Writing them this way makes it clear that this is functorial in the superpoints : given choices and , and any morphism between the corresponding and , 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 . (In particular, since , the ordinary point, is one of these, they have to commute with the projection to .) These conditions may be quite restrictive, but it leaves us with, at least, a quite concrete image of what maps of supermanifolds

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 ).

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 -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 , 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 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 , and the odd generators produce grade-reversing ones. (This fact that there are symmetries which flip the “bosonic” and “fermionic” parts of the total 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.

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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).

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 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 and are objects of , then there is an object , with an “evaluation map” , which makes it possible to think of as the object of “morphisms from A to B”.

The other main thing a topos has is a “subobject classifier”. Now, a subobject of is an equivalence class of monomorphisms into – 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 , whose elements we can tall “true” and “false”. Then every subset of corresponds to a characteristic function . In general, a subobject classifies is an object together with a map from the terminal object, , such that every inclusion of subobject is a pullback of 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 together with a “topology” , which is a rule which, for each , picks out , a set of *collections of maps* into , called *seives* for . They collections have to satisfy certain conditions, but the idea can be understood in terms of the basic example, . Given a topological space, is the category whose objects are the open sets , and the morphisms are all the inclusions. Then that each collection in is an open cover of – that is, a bunch of inclusions of open sets, which together cover all of in the usual sense.

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

The point is that a presheaf on is just a functor . That is, it’s a way of assigning a set to each . So, for instance, for either of the cases we just mentioned, one has , which assigns to each open set the set of all bounded functions on , and to every inclusion the restriction map. Or, again, one has , which assigns the set of all continuous functions.

These two examples illustrate the condition which distinguishes those presheaves which are *sheaves* – namely, those which satisfy some “gluing” conditions. Thus, suppose we’re, given an open cover , and a choice of one element from each , 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 which restricts to all the under the maps .

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 have corresponding objects and morphisms in . Namely, the objects appear as the representable presheaves, , and the morphisms show up as the induced natural transformations between these functors. This map is called the Yoneda embedding. If 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 lands in .

In this case, the Yoneda embedding embeds as a sub-category of . What’s more, it’s a full subcategory: all the natural transformations between representable presheaves come from the morphisms of -objects in a unique way. So is, in this sense, a generalization of itself.

More precisely, it’s the Yoneda lemma which makes sense of all this. The idea is to start with the way ordinary -objects (from now on, just call them “spaces”) become presheaves: they become functors which assign to each the set of all maps into . 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 , and any , the set is naturally isomorphic to : that is, literally is the collection of morphisms from (or rather, its image under the Yoneda embedding) and a “generalized space” . (See also Tom Leinster’s nice discussion of the Yoneda Lemma if this isn’t familiar.) We describe as a “probe” object: one probes the space by mapping into it in various ways. Knowing the results for all tells you all about the “space” . (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 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…

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 , for all choices of , 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 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 , then we have a slightly intuition: the “generalized” spaces are something like generalized bundles over , 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, -valued functions are sections of the bundle . Given a nontrivial bundle , there is a sheaf of sections – on each , one gets to be all the one-sided inverses which are one-sided inverses of . For a generic sheaf, we can imagine a sort of “generalized bundle” over .

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

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

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

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

The concrete case is therefore nice for getting an intuition for what it means to think of sheaves as spaces. For sheaves which aren’t concrete, morphisms aren’t determined by what they do to the underlying points i.e. the forgetful “underlying set” functor isn’t faithful. Here, we might think of a “generalized space” which looks like two copies of the same topological space: the sheaf gives two different elements of 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 – 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 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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As for last semester’s seminar, one of the two main threads, the one which Alessandro Valentino and I helped to organize, was a look at some of the material needed to approach Jacob Lurie’s paper on the classification of topological quantum field theories. The idea was for the research seminar to present the basic tools that are used in that paper to a larger audience, mostly of graduate students – enough to give a fairly precise statement, and develop the tools needed to follow the proof. (By the way, for a nice and lengthier discussion by Chris Schommer-Pries about this subject, which includes more details on much of what’s in this post, check out this video.)

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

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

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

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

.

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

The -category is equivalent to the free symmetric monoidal -category generated by one fully-dualizable object.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Now, if we want to build an -category of cobordisms, the idea is the same, except that we have a simplicial object, in a category of simplicial objects, and so on. However, the way to define this is essentially similar. To specify an -fold C.S.S., we have to specify a whole collection of spaces associated to cobordisms equipped with embeddings into . In particular, for each tuple , we have the space of such embeddings, such that for each one has special points along the coordinate axis. These are the ways of breaking down a given cobordism into a composite of pieces. Again, one has to make sure that these critical points of the Morse functions defined by the projections onto these coordinate axes avoid these special which define the manifolds where gluing takes place. The composition maps which make these into a simplical object are quite natural – they just come by deleting special points.

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

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

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

Now we want to classify what TQFT’s live on this category. These are functors . We have two generating objects, and , the two signed points. A TQFT must assign these objects vector spaces, which we’ll call and . Collections of points get assigned tensor products of all the corresponding vector spaces, since the functor is monoidal, so knowing these two vector spaces determines what does to all objects.

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

and a coevaluation:

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

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

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

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

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

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

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

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