2-groups


To continue from the previous post

Twisted Differential Cohomology

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

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

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

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

Fusion Categories

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

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

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

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

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

Quantizing with Higher Categories

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

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

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

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

A \leftarrow X \rightarrow B

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

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

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

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

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

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

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

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

TQFTs with Boundary

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

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

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

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

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

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

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

2-Knots

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

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

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

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

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

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

(…To be continued in Part 2…)

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

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

Global and Local Symmetry

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

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

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

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

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

Transformation Groupoid

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

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

actionsquare

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

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

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

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

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

Categorify Everything

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

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

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

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

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

2-Group Actions

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

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

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

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

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

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

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

The Transformation Double Category of a 2-Group Action

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

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

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

squarepic

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

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

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

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

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

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

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

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

Speculative Remarks

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

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

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

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

Hamburg

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

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

Brno Visit

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

This fellow was near the hotel I stayed in:

Image

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

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

Image

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

Image

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

Moduli Spaces in Higher Gauge Theory

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

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

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

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

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

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

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

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

2-Group Actions and the Transformation Double Category

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

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

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

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

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

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

pullback

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

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

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

action

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

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

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

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

squares

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

The horizontal arrow on the bottom of this square is:

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

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

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

Higher Symmetry of the Moduli Space

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

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

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

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

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

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

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

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

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

Well, as promised in the previous post, I’d like to give a summary of some of what was discussed at the conference I attended (quite a while ago now, late last year) in Erlangen, Germany.  I was there also to visit Derek Wise, talking about a project we’ve been working on for some time.

(I’ve also significantly revised this paper about Extended TQFT since then, and it now includes some stuff which was the basis of my talk at Erlangen on cohomological twisting of the category Span(Gpd).  I’ll get to that in the next post.  Also coming up, I’ll be describing some new things I’ve given some talks about recently which relate the Baez-Dolan groupoidification program to Khovanov-Lauda categorification of algebras – at least in one example, hopefully in a way which will generalize nicely.)

In the meantime, there were a few themes at the conference which bear on the Extended TQFT project in various ways, so in this post I’ll describe some of them.  (This isn’t an exhaustive description of all the talks: just of a selection of illustrative ones.)


Categories with Structures

A few talks were mainly about facts regarding the sorts of categories which get used in field theory contexts.  One important type, for instance, are fusion categories is a monoidal category which is enriched in vector spaces, generated by simple objects, and some other properties: essentially, monoidal 2-vector spaces.  The basic example would be categories of representations (of groups, quantum groups, algebras, etc.), but fusion categories are an abstraction of (some of) their properties.  Many of the standard properties are described and proved in this paper by Etingof, Nikshych, and Ostrik, which also poses one of the basic conjectures, the “ENO Conjecture”, which was referred to repeatedly in various talks.  This is the guess that every fusion category can be given a “pivotal” structure: an isomorphism from Id to **.  It generalizes the theorem that there’s always such an isomorphism into ****.  More on this below.

Hendryk Pfeiffer talked about a combinatorial way to classify fusion categories in terms of certain graphs (see this paper here).  One way I understand this idea is to ask how much this sort of category really does generalize categories of representations, or actually comodules.  One starting point for this is the theorem that there’s a pair of functors between certain monoidal categories and weak Hopf algebras.  Specifically, the monoidal categories are (Cat \downarrow Vect)^{\otimes}, which consists of monoidal categories equipped with a forgetful functor into Vect.  Then from this one can get (via a coend), a weak Hopf algebra over the base field k(in the category WHA_k).  From a weak Hopf algebra H, one can get back such a category by taking all the modules of H.  These two processes form an adjunction: they’re not inverses, but we have maps between the two composites and the identity functors.

The new result Hendryk gave is that if we restrict our categories over Vect to be abelian, and the functors between them to be linear, faithful, and exact (that is, roughly, that we’re talking about concrete monoidal 2-vector spaces), then this adjunction is actually an equivalence: so essentially, all such categories C may as well be module categories for weak Hopf algebras.  Then he gave a characterization of these in terms of the “dimension graph” (in fact a quiver) for (C,M), where M is one of the monoidal generators of C.  The vertices of \mathcal{G} = \mathcal{G}_{(C,M)} are labelled by the irreducible representations v_i (i.e. set of generators of the category), and there’s a set of edges j \rightarrow l labelled by a basis of Hom(v_j, v_l \otimes M).  Then one can carry on and build a big graded algebra H[\mathcal{G}] whose m-graded part consists of length-m paths in \mathcal{G}.  Then the point is that the weak Hopf algebra of which C is (up to isomorphism) the module category will be a certain quotient of H[\mathcal{G}] (after imposing some natural relations in a systematic way).

The point, then, is that the sort of categories mostly used in this area can be taken to be representation categories, but in general only of these weak Hopf algebras: groups and ordinary algebras are special cases, but they show up naturally for certain kinds of field theory.

Tensor Categories and Field Theories

There were several talks about the relationship between tensor categories of various sorts and particular field theories.  The idea is that local field theories can be broken down in terms of some kind of n-category: n-dimensional regions get labelled by categories, (n-1)-D boundaries between regions, or “defects”, are labelled by functors between the categories (with the idea that this shows how two different kinds of field can couple together at the defect), and so on (I think the highest-dimension that was discussed explicitly involved 3-categories, so one has junctions between defects, and junctions between junctions, which get assigned some higher-morphism data).  Alteratively, there’s the dual picture where categories are assigned to points, functors to 1-manifolds, and so on.  (This is just Poincaré duality in the case where the manifolds come with a decomposition into cells, which they often are if only for convenience).

Victor Ostrik gave a pair of talks giving an overview role tensor categories play in conformal field theory.  There’s too much material here to easily summarize, but the basics go like this: CFTs are field theories defined on cobordisms that have some conformal structure (i.e. notion of angles, but not distance), and on the algebraic side they are associated with vertex algebras (some useful discussion appears on mathoverflow, but in this context they can be understood as vector spaces equipped with exactly the algebraic operations needed to model cobordisms with some local holomorphic structure).

In particular, the irreducible representations of these VOA’s determine the “conformal blocks” of the theory, which tell us about possible correlations between observables (self-adjoint operators).  A VOA V is “rational” if the category Rep(V) is semisimple (i.e. generated as finite direct sums of these conformal blocks).  For good VOA’s, Rep(V) will be a modular tensor category (MTC), which is a fusion category with a duality, braiding, and some other strucutre (see this for more).   So describing these gives us a lot of information about what CFT’s are possible.

The full data of a rational CFT are given by a vertex algebra, and a module category M: that is, a fusion category is a sort of categorified ring, so it can act on M as an ring acts on a module.  It turns out that choosing an M is equivalent to finding a certain algebra (i.e. algebra object) \mathcal{L}, a “Lagrangian algebra” inside the centre of Rep(V).  The Drinfel’d centre Z(C) of a monoidal category C is a sort of free way to turn a monoidal category into a braided one: but concretely in this case it just looks like Rep(V) \otimes Rep(V)^{\ast}.  Knowing the isomorphism class \mathcal{L} determines a “modular invariant”.  It gets “physics” meaning from how it’s equipped with an algebra structure (which can happen in more than one way), but in any case \mathcal{L} has an underlying vector space, which becomes the Hilbert space of states for the conformal field theory, which the VOA acts on in the natural way.

Now, that was all conformal field theory.  Christopher Douglas described some work with Chris Schommer-Pries and Noah Snyder about fusion categories and structured topological field theories.  These are functors out of cobordism categories, the most important of which are n-categories, where the objects are points, morphisms are 1D cobordisms, and so on up to n-morphisms which are n-dimensional cobordisms.  To keep things under control, Chris Douglas talked about the case Bord_0^3, which is where n=3, and a “local” field theory is a 3-functor Bord_0^3 \rightarrow \mathcal{C} for some 3-category \mathcal{C}.  Now, the (Baez-Dolan) Cobordism Hypothesis, which was proved by Jacob Lurie, says that Bord_0^3 is, in a suitable sense, the free symmetric monoidal 3-category with duals.  What this amounts to is that a local field theory whose target 3-category is \mathcal{C} is “just” a dualizable object of \mathcal{C}.

The handy example which links this up to the above is when \mathcal{C} has objects which are tensor categories, morphisms which are bimodule categories (i.e. categories acted), 2-morphisms which are functors, and 3-morphisms which are natural transformations.  Then the issue is to classify what kind of tensor categories these objects can be.

The story is trickier if we’re talking about, not just topological cobordisms, but ones equipped with some kind of structure regulated by a structure group G(for instance, orientation by G=SO(n), spin structure by its universal cover G= Spin(n), and so on).  This means the cobordisms come equipped with a map into BG.  They take O(n) as the starting point, and then consider groups G with a map to O(n), and require that the map into BG is a lift of the map to BO(n).  Then one gets that a structured local field theory amounts to a dualizable objects of \mathcal{C} with a homotopy-fixed point for some G-action – and this describes what gets assigned to the point by such a field theory.  What they then show is a correspondence between G and classes of categories.  For instance, fusion categories are what one gets by imposing that the cobordisms be oriented.

Liang Kong talked about “Topological Orders and Tensor Categories”, which used the Levin-Wen models, from condensed matter phyiscs.  (Benjamin Balsam also gave a nice talk describing these models and showing how they’re equivalent to the Turaev-Viro and Kitaev models in appropriate cases.  Ingo Runkel gave a related talk about topological field theories with “domain walls”.).  Here, the idea of a “defect” (and topological order) can be understood very graphically: we imagine a 2-dimensional crystal lattice (of atoms, say), and the defect is a 1-dimensional place where the two lattices join together, with the internal symmetry of each breaking down at the boundary.  (For example, a square lattice glued where the edges on one side are offset and meet the squares on the other side in the middle of a face, as you typically see in a row of bricks – the slides linked above have some pictures).  The Levin-Wen models are built using a hexagonal lattice, starting with a tensor category with several properties: spherical (there are dualities satisfying some relations), fusion, and unitary: in fact, historically, these defining properties were rediscovered independently here as the requirement for there to be excitations on the boundary which satisfy physically-inspired consistency conditions.

These abstract the properties of a category of representations.  A generalization of this to “topological orders” in 3D or higher is an extended TFT in the sense mentioned just above: they have a target 3-category of tensor categories, bimodule categories, functors and natural transformations.  The tensor categories (say, \mathcal{C}, \mathcal{D}, etc.) get assigned to the bulk regions; to “domain walls” between different regions, namely defects between lattices, we assign bimodule categories (but, for instance, to a line within a region, we get \mathcal{C} understood as a \mathcal{C}-\mathcal{C}-bimodule); then to codimension 2 and 3 defects we attach functors and natural transformations.  The algebra for how these combine expresses the ways these topological defects can go together.  On a lattice, this is an abstraction of a spin network model, where typically we have just one tensor category \mathcal{C} applied to the whole bulk, namely the representations of a Lie group (say, a unitary group).  Then we do calculations by breaking down into bases: on codimension-1 faces, these are simple objects of \mathcal{C}; to vertices we assign a Hom space (and label by a basis for intertwiners in the special case); and so on.

Thomas Nickolaus spoke about the same kind of G-equivariant Dijkgraaf-Witten models as at our workshop in Lisbon, so I’ll refer you back to my earlier post on that.  However, speaking of equivariance and group actions:

Michael Müger  spoke about “Orbifolds of Rational CFT’s and Braided Crossed G-Categories” (see this paper for details).  This starts with that correspondence between rational CFT’s (strictly, rational chiral CFT’s) and modular categories Rep(F).  (He takes F to be the name of the CFT).  Then we consider what happens if some finite group G acts on F (if we understand F as a functor, this is an action by natural transformations; if as an algebra, then ).  This produces an “orbifold theory” F^G (just like a finite group action on a manifold produces an orbifold), which is the “G-fixed subtheory” of F, by taking G-fixed points for every object, and is also a rational CFT.  But that means it corresponds to some other modular category Rep(F^G), so one would like to know what category this is.

A natural guess might be that it’s Rep(F)^G, where C^G is a “weak fixed-point” category that comes from a weak group action on a category C.  Objects of C^G are pairs (c,f_g) where c \in C and f_g : g(c) \rightarrow c is a specified isomorphism.  (This is a weak analog of S^G, the set of fixed points for a group acting on a set).  But this guess is wrong – indeed, it turns out these categories have the wrong dimension (which is defined because the modular category has a trace, which we can sum over generating objects).  Instead, the right answer, denoted by Rep(F^G) = G-Rep(F)^G, is the G-fixed part of some other category.  It’s a braided crossed G-category: one with a grading by G, and a G-action that gets along with it.  The identity-graded part of Rep(F^G) is just the original Rep(F).

State Sum Models

This ties in somewhat with at least some of the models in the previous section.  Some of these were somewhat introductory, since many of the people at the conference were coming from a different background.  So, for instance, to begin the workshop, John Barrett gave a talk about categories and quantum gravity, which started by outlining the historical background, and the development of state-sum models.  He gave a second talk where he began to relate this to diagrams in Gray-categories (something he also talked about here in Lisbon in February, which I wrote about then).  He finished up with some discussion of spherical categories (and in particular the fact that there is a Gray-category of spherical categories, with a bunch of duals in the suitable sense).  This relates back to the kind of structures Chris Douglas spoke about (described above, but chronologically right after John).  Likewise, Winston Fairbairn gave a talk about state sum models in 3D quantum gravity – the Ponzano Regge model and Turaev-Viro model being the focal point, describing how these work and how they’re constructed.  Part of the point is that one would like to see that these fit into the sort of framework described in the section above, which for PR and TV models makes sense, but for the fancier state-sum models in higher dimensions, this becomes more complicated.

Higher Gauge Theory

There wasn’t as much on this topic as at our own workshop in Lisbon (though I have more remarks on higher gauge theory in one post about it), but there were a few entries.  Roger Picken talked about some work with Joao Martins about a cubical formalism for parallel transport based on crossed modules, which consist of a group G and abelian group H, with a map \partial : H \rightarrow G and an action of G on H satisfying some axioms.  They can represent categorical groups, namely group objects in Cat (equivalently, categories internal to Grp), and are “higher” analogs of groups with a set of elements.  Roger’s talk was about how to understand holonomies and parallel transports in this context.  So, a “connection” lets on transport things with G-symmetries along paths, and with H-symmetries along surfaces.  It’s natural to describe this with squares whose edges are labelled by G-elements, and faces labelled by H-elements (which are the holonomies).  Then the “cubical approach” means that we can describe gauge transformations, and higher gauge transformations (which in one sense are the point of higher gauge theory) in just the same way: a gauge transformation which assigns H-values to edges and G-values to vertices can be drawn via the holonomies of a connection on a cube which extends the original square into 3D (so the edges become squares, and so get H-values, and so on).  The higher gauge transformations work in a similar way.  This cubical picture gives a good way to understand the algebra of how gauge transformations etc. work: so for instance, gauge transformations look like “conjugation” of a square by four other squares – namely, relating the front and back faces of a cube by means of the remaining faces.  Higher gauge transformations can be described by means of a 4D hypercube in an analogous way, and their algebraic properties have to do with the 2D faces of the hypercube.

Derek Wise gave a short talk outlining his recent paper with John Baez in which they show that it’s possible to construct a higher gauge theory based on the Poincare 2-group which turns out to have fields, and dynamics, which are equivalent to teleparallel gravity, a slightly unusal theory which nevertheless looks in practice just like General Relativity.  I discussed this in a previous post.

So next time I’ll talk about the new additions to my paper on ETQFT which were the basis of my talk, which illustrates a few of the themes above.

(Note: WordPress seems to be having some intermittent technical problem parsing my math markup in this post, so please bear with me until it, hopefully, goes away…)

As August is the month in which Portugal goes on vacation, and we had several family visitors toward the end of the summer, I haven’t posted in a while, but the term has now started up at IST, and seminars are underway, so there should be some interesting stuff coming up to talk about.

New Blog

First, I’ll point out that that Derek Wise has started a new blog, called simply “Simplicity“, which is (I imagine) what it aims to contain: things which seem complex explained so as to reveal their simplicity.  Unless I’m reading too much into the title.  As of this writing, he’s posted only one entry, but a lengthy one that gives a nice explanation of a program for categorified Klein geometries which he’s been thinking a bunch about.  Klein’s program for describing the geometry of homogeneous spaces (such as spherical, Euclidean, and hyperbolic spaces with constant curvature, for example) was developed at Erlangen, and goes by the name “The Erlangen Program”.  Since Derek is now doing a postdoc at Erlangen, and this is supposed to be a categorification of Klein’s approach, he’s referred to it the “2-Erlangen Program”.  There’s more discussion about it in a (somewhat) recent post by John Baez at the n-Category Cafe.  Both of them note the recent draft paper they did relating a higher gauge theory based on the Poincare 2-group to a theory known as teleparallel gravity.  I don’t know this theory so well, except that it’s some almost-equivalent way of formulating General Relativity

I’ll refer you to Derek’s own post for full details of what’s going on in this approach, but the basic motivation isn’t too hard to set out.  The Erlangen program takes the view that a homogeneous space is a space X (let’s say we mean by this a topological space) which “looks the same everywhere”.  More precisely, there’s a group action by some G, which we understand to be “symmetries” of the space, which is transitive.  Since every point is taken to every other point by some symmetry, the space is “homogeneous”.  Some symmetries leave certain points x \in X where they are – they form the stabilizer subgroup H = Stab(x).  When the space is homogeneous, it is isomorphic to the coset space, X \cong G / H.  So Klein’s idea is to say that any time you have a Lie group G and a closed subgroup H < G, this quotient will be called a “homogeneous space”.  A familiar example would be Euclidean space, \mathbb{R}^n \cong E(n) / O(n), where E is the Euclidean group and O is the orthogonal group, but there are plenty of others.

This example indicates what Cartan geometry is all about, though – this is the next natural step after Klein geometry (Edit:  Derek’s blog now has a visual explanation of Cartan geometry, a.k.a. “generalized hamsterology”, new since I originally posted this).  We can say that Cartan is to Klein as Riemann is to Euclid.  (Or that Cartan is to Riemann as Klein is to Euclid – or if you want to get maybe too-precisely metaphorical, Cartan is the pushout of Klein and Riemann over Euclid).  The point is that Riemannian geometry studies manifolds – spaces which are not homogeneous, but look like Euclidean space locally.  Cartan geometry studies spaces which aren’t homogeneous, but can be locally modelled by Klein geometries.  Now, a Riemannian geometry is essentially a manifold with a metric, describing how it locally looks like Euclidean space.  An equivalent way to talk about it is a manifold with a bundle of Euclidean spaces (the tangent spaces) with a connection (the Levi-Civita connection associated to the metric).  A Cartan geometry can likewise be described as a G-bundle with fibre X with a connection

Then the point of the “2-Erlangen program” is to develop similar geometric machinery for 2-groups (a.k.a. categorical groups).  This is, as usual, a bit more complicated since actions of 2-groups are trickier than group-actions.  In their paper, though, the point is to look at spaces which are locally modelled by some sort of 2-Klein geometry which derives from the Poincare 2-group.  By analogy with Cartan geometry, one can talk about such Poincare 2-group connections on a space – that is, some kind of “higher gauge theory”.  This is the sort of framework where John and Derek’s draft paper formulates teleparallel gravity.  It turns out that the 2-group connection ends up looking like a regular connection with torsion, and this plays a role in that theory.  Their draft will give you a lot more detail.

Talk on Manifold Calculus

On a different note, one of the first talks I went to so far this semester was one by Pedro Brito about “Manifold Calculus and Operads” (though he ran out of time in the seminar before getting to talk about the connection to operads).  This was about motivating and introducing the Goodwillie Calculus for functors between categories of spaces.  (There are various references on this, but see for instance these notes by Hal Sadofsky). In some sense this is a generalization of calculus from functions to functors, and one of the main results Goodwillie introduced with this subject, is a functorial analog of Taylor’s theorem.  I’d seen some of this before, but this talk was a nice and accessible intro to the topic.

So the starting point for this “Manifold Calculus” is that we’d like to study functors from spaces to spaces (in fact this all applies to spectra, which are more general, but Pedro Brito’s talk was focused on spaces).  The sort of thing we’re talking about is a functor which, given a space M, gives a moduli space of some sort of geometric structures we can put on M, or of mappings from M.  The main motivating example he gave was the functor

Imm(-,N) : [Spaces] \rightarrow [Spaces]

for some fixed manifold N. Given a manifold M, this gives the mapping space of all immersions of M into N.

(Recalling some terminology: immersions are maps of manifolds where the differential is nondegenerate – the induced map of tangent spaces is everywhere injective, meaning essentially that there are no points, cusps, or kinks in the image, but there might be self-intersections. Embeddings are, in addition, local homeomorphisms.)

Studying this functor Imm(-,N) means, among other things, looking at the various spaces Imm(M,N) of immersions of each M into N. We might first ask: can M be immersed in N at all – in other words, is \pi_0(Imm(M,N)) nonempty?

So, for example, the Whitney Embedding Theorem says that if dim(N) is at least 2 dim(M), then there is an embedding of M into N (which is therefore also an immersion).

In more detail, we might want to know what \pi_0(Imm(M,N)) is, which tells how many connected components of immersions there are: in other words, distinct classes of immersions which can’t be deformed into one another by a family of immersions. Or, indeed, we might ask about all the homotopy groups of Imm(M,N), not just the zeroth: what’s the homotopy type of Imm(M,N)? (Once we have a handle on this, we would then want to vary M).

It turns out this question is manageable, party due to a theorem of Smale and Hirsch, which is a generalization of Gromov’s h-principle – the original principle applies to solutions of certain kinds of PDE’s, saying that any solution can be deformed to a holomorphic one, so if you want to study the space of solutions up to homotopy, you may as well just study the holomorphic solutions.

The Smale-Hirsch theorem likewise gives a homotopy equivalence of two spaces, one of which is Imm(M,N). The other is the space of “formal immersions”, called Imm^f(M,N). It consists of all (f,F), where f : M \rightarrow N is smooth, and F : TM \rightarrow TN is a map of tangent spaces which restricts to f, and is injective. These are “formally” like immersions, and indeed Imm(M,N) has an inclusion into Imm^f(M,N), which happens to be a homotopy equivalence: it induces isomorphisms of all the homotopy groups. These come from homotopies taking each “formal immersion” to some actual immersion. So we’ve approximated Imm(-,N), up to homotopy, by Imm^f(-,N). (This “homotopy” of functors makes sense because we’re talking about an enriched functor – the source and target categories are enriched in spaces, where the concepts of homotopy theory are all available).

We still haven’t got to manifold calculus, but it will be all about approximating one functor by another – or rather, by a chain of functors which are supposed to be like the Taylor series for a function. The way to get this series has to do with sheafification, so first it’s handy to re-describe what the Smale-Hirsch theorem says in terms of sheaves. This means we want to talk about some category of spaces with a Grothendieck topology.

So lets let \mathcal{E} be the category whose objects are d-dimensional manifolds and whose morphisms are embeddings (which, of course, are necessarily codimension 0). Now, the point here is that if f : M \rightarrow M' is an embedding in \mathcal{E}, and M' has an immersion into N, this induces an immersion of M into N. This amounst to saying Imm(-,N) is a contravariant functor:

Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces]

That makes Imm(-,N) a presheaf. What the Smale-Hirsch theorem tells us is that this presheaf is a homotopy sheaf – but to understand that, we need a few things first.

First, what’s a homotopy sheaf? Well, the condition for a sheaf says that if we have an open cover of M, then

So to say how Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces] is a homotopy sheaf, we have to give \mathcal{E} a topology, which means defining a “cover”, which we do in the obvious way – a cover is a collection of morphisms f_i : U_i \rightarrow M such that the union of all the images \cup f_i(U_i) is just M. The topology where this is the definition of a cover can be called J_1, because it has the property that given any open cover and choice of 1 point in M, that point will be in some U_i of the cover.

This is part of a family of topologies, where J_k only allows those covers with the property that given any choice of k points in M, some open set of the cover contains them all. These conditions, clearly, get increasingly restrictive, so we have a sequence of inclusions (a “filtration”):

J_1 \leftarrow J_2 \leftarrow J_3 \leftarrow \dots

Now, with respect to any given one of these topologies J_k, we have the usual situation relating sheaves and presheaves.  Sheaves are defined relative to a given topology (i.e. a notion of cover).  A presheaf on \mathcal{E} is just a contravariant functor from \mathcal{E} (in this case valued in spaces); a sheaf is one which satisfies a descent condition (I’ve discussed this before, for instance here, when I was running the Stacks Seminar at UWO).  The point of a descent condition, for a given topology is that if we can take the values of a functor F “locally” – on the various objects of a cover for M – and “glue” them to find the value for M itself.  In particular, given a cover for M \in \mathcal{E}, and a cover, there’s a diagram consisting of the inclusions of all the double-overlaps of sets in the cover into the original sets.  Then the descent condition for sheaves of spaces is that

The general fact is that there’s a reflective inclusion of sheaves into presheaves (see some discussion about reflective inclusions, also in an earlier post).  Any sheaf is a contravariant functor – this is the inclusion of Sh( \mathcal{E} ) into $latex PSh( \mathcal{E} )$.  The reflection has a left adjoint, sheafification, which takes any presheaf in PSh( \mathcal{E} ) to a sheaf which is the “best approximation” to it.  It’s the fact this is an adjoint which makes the inclusion “reflective”, and provides the sense in which the sheafification is an approximation to the original functor.

The way sheafification works can be worked out from the fact that it’s an adjoint to the inclusion, but it also has a fairly concrete description.  Given any one of the topologies J_k,  we have a whole collection of special diagrams, such as:

U_i \leftarrow U_{ij} \rightarrow U_j

(using the usual notation where U_{ij} = U_i \cap U_j is the intersection of two sets in a cover, and the maps here are the inclusions of that intersection).  This and the various other diagrams involving these inclusions are special, given the topology J_k.  The descent condition for a sheaf F says that if we take the image of this diagram:

F(U_i) \rightarrow F(U_{ij}) \leftarrow F(U_j)

then we can “glue together” the objects F(U_i) and F(U_j) on the overlap to get one on the union.  That is, F is a sheaf if F(U_i \cup U_j) is a colimit of the diagram above (intuitively, by “gluing on the overlap”).  In a presheaf, it would come equipped with some maps into the F(U_i) and F(U_j): in a sheaf, this object and the maps satisfy some universal property.  Sheafification takes a presheaf F to a sheaf F^{(k)} which does this, essentially by taking all these colimits.  More accurately, since these sheaves are valued in spaces, what we really want are homotopy sheaves, where we can replace “colimit” with “homotopy colimit” in the above – which satisfies a universal property only up to homotopy, and which has a slightly weaker notion of “gluing”.   This (homotopy) sheaf is called F^{(k)} because it depends on the topology J_k which we were using to get the class of special diagrams.

One way to think about F^{(k)} is that we take the restriction to manifolds which are made by pasting together at most k open balls.  Then, knowing only this part of the functor F, we extend it back to all manifolds by a Kan extension (this is the technical sense in which it’s a “best approximation”).

Now the point of all this is that we’re building a tower of functors that are “approximately” like F, agreeing on ever-more-complicated manifolds, which in our motivating example is F = Imm(-,N).  Whichever functor we use, we get a tower of functors connected by natural transformations:

F^{(1)} \leftarrow F^{(2)} \leftarrow F^{(3)} \leftarrow \dots

This happens because we had that chain of inclusions of the topologies J_k.  Now the idea is that if we start with a reasonably nice functor (like F = Imm(-,N) for example), then F is just the limit of this diagram.  That is, it’s the universal thing F which has a map into each F^{(k)} commuting with all these connecting maps in the tower.  The tower of approximations – along with its limit (as a diagram in the category of functors) – is what Goodwillie called the “Taylor tower” for F.  Then we say the functor F is analytic if it’s just (up to homotopy!) the limit of this tower.

By analogy, think of an inclusion of a vector space V with inner product into another such space W which has higher dimension.  Then there’s an orthogonal projection onto the smaller space, which is an adjoint (as a map of inner product spaces) to the inclusion – so these are like our reflective inclusions.  So the smaller space can “reflect” the bigger one, while not being able to capture anything in the orthogonal complement.  Now suppose we have a tower of inclusions V \leftarrow V' \leftarrow V'' \dots, where each space is of higher dimension, such that each of the V is included into W in a way that agrees with their maps to each other.  Then given a vector w \in W, we can take a sequence of approximations (v,v',v'',\dots) in the V spaces.  If w was “nice” to begin with, this series of approximations will eventually at least converge to it – but it may be that our tower of V spaces doesn’t let us approximate every w in this way.

That’s precisely what one does in calculus with Taylor series: we have a big vector space W of smooth functions, and a tower of spaces we use to approximate.  These are polynomial functions of different degrees: first linear, then quadratic, and so forth.  The approximations to a function f are orthogonal projections onto these smaller spaces.  The sequence of approximations, or rather its limit (as a sequence in the inner product space W), is just what we mean by a “Taylor series for f“.  If f is analytic in the first place, then this sequence will converge to it.

The same sort of phenomenon is happening with the Goodwillie calculus for functors: our tower of sheafifications of some functor F are just “projections” onto smaller categories (of sheaves) inside the category of all contravariant functors.  (Actually, “reflections”, via the reflective inclusions of the sheaf categories for each of the topologies J_k).  The Taylor Tower for this functor is just like the Taylor series approximating a function.  Indeed, this analogy is fairly close, since the topologies J_k will give approximations of F which are in some sense based on k points (so-called k-excisive functors, which in our terminology here are sheaves in these topologies).  Likewise, a degree-k polynomial approximation approximates a smooth function, in general in a way that can be made to agree at k points.

Finally, I’ll point out that I mentioned that the Goodwillie calculus is actually more general than this, and applies not only to spaces but to spectra. The point is that the functor Imm(-,N) defines a kind of generalized cohomology theory – the cohomology groups for M are the \pi_i(Imm(M,N)). So the point is, functors satisfying the axioms of a generalized cohomology theory are represented by spectra, whereas N here is a special case that happens to be a space.

Lots of geometric problems can be thought of as classified by this sort of functor – if N = BG, the classifying space of a group, and we drop the requirement that the map be an immersion, then we’re looking at the functor that gives the moduli space of G-connections on each M.  The point is that the Goodwillie calculus gives a sense in which we can understand such functors by simpler approximations to them.

Continuing from the previous post, there are a few more lecture series from the school to talk about.

Higher Gauge Theory

The next was John Huerta’s series on Higher Gauge Theory from the point of view of 2-groups.  John set this in the context of “categorification”, a slightly vague program of replacing set-based mathematical ideas with category-based mathematical ideas.  The general reason for this is to get an extra layer of “maps between things”, or “relations between relations”, etc. which tend to be expressed by natural transformations.  There are various ways to go about this, but one is internalization: given some sort of structure, the relevant example in this case being “groups”, one has a category {Groups}, and can define a 2-group as a “category internal to {Groups}“.  So a 2-group has a group of objects, a group of morphisms, and all the usual maps (source and target for morphisms, composition, etc.) which all have to be group homomorphisms.  It should be said that this all produces a “strict 2-group”, since the objects G necessarily form a group here.  In particular, m : G \times G \rightarrow G satisfies group axioms “on the nose” – which is the only way to satisfy them for a group made of the elements of a set, but for a group made of the elements of a category, one might require only that it commute up to isomorphism.  A weak 2-group might then be described as a “weak model” of the theory of groups in Cat, but this whole approach is much less well-understood than the strict version as one goes to general n-groups.

Now, as mentioned in the previous post, there is a 1-1 correspondence between 2-groups and crossed modules (up to equivalence): given a crossed module (G,H,\partial,\rhd), there’s a 2-group \mathcal{G} whose objects are G and whose morphisms are G \ltimes H; given a 2-group \mathcal{G} with objects G, there’s a crossed module (G, Aut(1_G),1,m).  (The action m acts on a morphism in such as way as to act by multiplication on its source and target).  Then, for instance, the Peiffer identity for crossed modules (see previous post) is a consequence of the fact that composition of morphisms is supposed to be a group homomorphism.

Looking at internal categories in [your favourite setting here] isn’t the only way to do categorification, but it does produce some interesting examples.  Baez-Crans 2-vector spaces are defined this way (in Vect), and built using these are Lie 2-algebras.  Looking for a way to integrate Lie 2-algebras up to Lie 2-groups (which are internal categories in Lie groups) brings us back to the current main point.  This is the use of 2-groups to do higher gauge theory.  This requires the use of “2-bundles”.  To explain these, we can say first of all that a “2-space” is an internal category in Spaces (whether that be manifolds, or topological spaces, or what-have-you), and that a (locally trivial) 2-bundle should have a total 2-space E, a base 2-space M, and a (functorial) projection map p : E \rightarrow M, such that there’s some open cover of M by neighborhoods U_i where locally the bundle “looks like” \pi_i : U_i \times F \rightarrow U_i, where F is the fibre of the bundle.  In the bundle setting, “looks like” means “is isomorphic to” by means of isomorphisms f_i : E_{U_i} \rightarrow U_i \times F.  With 2-bundles, it’s interpreted as “is equivalent to” in the categorical sense, likewise by maps f_i.

Actually making this precise is a lot of work when M is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard.  This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where M is just an ordinary manifold (thought of as a 2-space which has only identity morphisms).   Then a key point is that there’s an analog to how (principal) G-bundles (where F \cong G as a G-set) are classified up to isomorphism by the first Cech cohomology of the manifold, \check{H}^1(M,G).  This works because one can define transition functions on double overlaps U_{ij} := U_i \cap U_j, by g_{ij} = f_i f_j^{-1}.  Then these g_{ij} will automatically satisfy the 1-cocycle condidion (g_{ij} g_{jk} = g_{ik} on the triple overlap U_{ijk}) which means they represent a cohomology class [g] = \in \check{H}^1(M,G).

A comparable thing can be said for the “transition functors” for a 2-bundle – they’re defined superficially just as above, except that being functors, we can now say there’s a natural isomorphism h_{ijk} : g_{ij}g_{jk} \rightarrow g_{ik}, and it’s these h_{ijk}, defined on triple overlaps, which satisfy a 2-cocycle condition on 4-fold intersections (essentially, the two ways to compose them to collapse g_{ij} g_{jk} g_{kl} into g_{il} agree).  That is, we have g_{ij} : U_{ij} \rightarrow Ob(\mathcal{G}) and h_{ijk} : U_{ijk} \rightarrow Mor(\mathcal{G}) which fit together nicely.  In particular, we have an element [h] \in \check{H}^2(M,G) of the second Cech cohomology of M: “principal \mathcal{G}-bundles are classified by second Cech cohomology of M“.  This sort of thing ties in to an ongoing theme of the later talks, the relationship between gerbes and higher cohomology – a 2-bundle corresponds to a “gerbe”, or rather a “1-gerbe”.  (The consistent terminology would have called a bundle a “0-gerbe”, but as usual, terminology got settled before the general pattern was understood).

Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles.  A connection on a bundle gives a parallel transport operation for paths \gamma in M, telling how to identify the fibres at points along \gamma by means of a functor hol : P_1(M) \rightarrow G, thinking of G as a category with one object, and where P_1(M) is the path groupoid whose objects are points in M and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around \gamma, anyway, to think of it as M \times G locally – in general we need to get the transition functions involved when we pass into some other local neighborhood.  A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of hol : P_2(M) \rightarrow \mathcal{G}, where \mathcal{G} is seen as a 2-category with one object, and P_2(M) now has 2-morphisms which are (essentially) homotopies of paths.

Just as connections can be described by 1-forms A valued in Lie(G), which give hol by integrating, a similar story exists for 2-connections: now we need a 1-form A valued in Lie(G) and a 2-form B valued in Lie(H).  These need to satisfy some relations, essentially that the curvature of A has to be controlled by B.   Moreover, that B is related to the B-field of string theory, as I mentioned in the post on the pre-school… But really, this is telling us about the Lie 2-algebra associated to \mathcal{G}, and how to integrate it up to the group!

Infinite Dimensional Lie Theory and Higher Gauge Theory

This series of talks by Christoph Wockel returns us to the question of “integrating up” to a Lie group G from a Lie algebra \mathfrak{g} = Lie(G), which is seen as the tangent space of G at the identity.  This is a well-understood, well-behaved phenomenon when the Lie algebras happen to be finite dimensional.  Indeed the classification theorem for the classical Lie groups can be got at in just this way: a combinatorial way to characterize Lie algebras using Dynkin diagrams (which describe the structure of some weight lattice), followed by a correspondence between Lie algebras and Lie groups.  But when the Lie algebras are infinite dimensional, this just doesn’t have to work.  It may be impossible to integrate a Lie algebra up to a full Lie group: instead, one can only get a little neighborhood of the identity.  The point of such infinite-dimensional groups, and ultimately their representation theory, is to deal with string groups that have to do with motions of extended objects.  Christoph Wockel was describing a result which says that, going to 2-groups, this problem can be overcome.  (See the relevant paper here.)

The first lecture in the series presented some background on a setting for infinite dimensional manifolds.  There are various approaches, a popular one being Frechet manifolds, but in this context, the somewhat weaker notion of locally convex spaces is sufficient.  These are “locally modelled” by (infinite dimensional) locally convex vector spaces, the way finite dimensonal manifolds are locally modelled by Euclidean space.  Being locally convex is enough to allow them to support a lot of differential calculus: one can find straight-line paths, locally, to define a notion of directional derivative in the direction of a general vector.  Using this, one can build up definitions of differentiable and smooth functions, derivatives, and integrals, just by looking at the restrictions to all such directions.  Then there’s a fundamental theorem of calculus, a chain rule, and so on.

At this point, one has plenty of differential calculus, and it becomes interesting to bring in Lie theory.  A Lie group is defined as a group object in the category of manifolds and smooth maps, just as in the finite-dimensional case.  Some infinite-dimensional Lie groups of interest would include: G = Diff(M), the group of diffeomorphisms of some compact manifold M; and the group of smooth functions G = C^{\infty}(M,K) from M into some (finite-dimensional) Lie group K (perhaps just \mathbb{R}), with the usual pointwise multiplication.  These are certainly groups, and one handy fact about such groups is that, if they have a manifold structure near the identity, on some subset that generates G as a group in a nice way, you can extend the manifold structure to the whole group.  And indeed, that happens in these examples.

Well, next we’d like to know if we can, given an infinite dimensional Lie algebra X, “integrate up” to a Lie group – that is, find a Lie group G for which X \cong T_eG is the “infinitesimal” version of G.  One way this arises is from central extensions.  A central extension of Lie group G by Z is an exact sequence Z \hookrightarrow \hat{G} \twoheadrightarrow G where (the image of) Z is in the centre of \hat{G}.  The point here is that \hat{G} extends G.  This setup makes \hat{G} is a principal Z-bundle over G.

Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps.  There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map.  The question is to go the other way: given one of these, get back to an extension of Lie groups.  The problem of finding extensions of G by Z, in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory.  One type of extension is the universal cover of G, which appears as \pi_1(G) \hookrightarrow \hat{G} \twoheadrightarrow G, so that the fibre is \pi_1(G).

In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function f : G \times G \rightarrow Z which is locally smooth (i.e. in some neighborhood in G), and is a cocyle (so that f(g,h) + f(gh,k) = f(g,hk) + f(h,k)), by the same sorts of arguments we’ve already seen a bit of.  For this reason, central extensions are classified by the cohomology group H^2(G,Z).  The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in G, and is used to construct \hat{G} (by specifying how multiplication on G lifts to \hat{G}).  Given a  2-cocycle \omega at the Lie algebra level (easier to do), one would like to lift that up the Lie group.  It turns out this is possible if the period homomorphism per_{\omega} : \Pi_2(G) \rightarrow Z – which takes a chain [\sigma] (with \sigma : S^2 \rightarrow G) to the integral of the original cocycle on it, \int_{\sigma} \omega – lands in a discrete subgroup of Z. A popular example of this is when Z is just \mathbb{R}, and the discrete subgroup is \mathbb{Z} (or, similarly, U(1) and 1 respectively).  This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.

So suppose we wanted to make the “2-connected cover” \pi_2(G) \hookrightarrow \pi_2(G) \times_{\gamma} G \twoheadrightarrow G as a central extension: since \pi_2(G) will be abelian, this is conceivable.  If the dimension of G is finite, this is trivial (since \pi_2(G) = 0 in finite dimensions), which is why we need some theory  of infinite-dimensional manifolds.  Moreover, though, this may not work in the context of groups: the \gamma in the extension \pi_2(G) \times_{\gamma} G above needs to be a “twisting” of associativity, not multiplication, being lifted from G.  Such twistings come from the THIRD cohomology of G (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like).  In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere).  Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case.  Then by thinking of \pi_2(G) and G as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of G as the base group of the crossed module, and \pi_2(G) as the second group in the tower.

The pattern of moving to higher group-like structures, higher cohomology, and obstructions to various constructions ran all through the workshop, and carried on in the next school session…

Higher Spin Structures in String Theory

Hisham Sati gave just one school-lecture in addition to his workshop talk, but it was packed with a lot of material.  This is essentially about cohomology and the structures on manifolds to which cohomology groups describe the obstructions.  The background part of the lecture referenced this book by Fridrich, and the newer parts were describing some of Sati’s own work, in particular a couple of papers with Schreiber and Stasheff (also see this one).

The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle.  For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives.  These two structures really belong on vector bundles or spinor bundles.  Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation.  So having nontrivial bundles really makes this all more interesting.  However, this isn’t always possible, and so one wants to the obstruction to being able to do it.  This is typically a class in one of the cohomology groups of the manifold – a characteristic class.  There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees i.  Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups U and O which are the limits of the unitary and orthogonal groups (such as O := O(\infty) \supset \dots \supset O(2) \supset O(1))

The point is that these classes are obstructions to building certain structures on the manifold X – which amounts to finding sections of a bundle.  So for instance, the first Steifel-Whitney classes, w_1(E) of a bundle E are related to orientations, coming from cohomology with coefficients in O(n).  Orientations for the manifold X can be described in terms of its tangent bundle, which is an O(n)-bundle (tangent spaces carry an action of the rotation group).  Consider X = S^1, where we have actually O(1) \simeq \mathbb{Z}_2.  The group H^1(S^1, \mathbb{Z}_2) has two elements, and there are two types of line bundle on the circle S^1: ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip.  The circle is orientable, because its tangent bundle is of the first sort.

Generally, an orientation can be put on X if the tangent bundle, as a map f : X \rightarrow B(O(n)), can be lifted to a map \tilde{f} : X \rightarrow B(SO(n)) – that is, it’s “secretly” an SO(n)-bundle – the special orthogonal group respects orientation, which is what the determinant measures.  Its two values, \pm 1, are what’s behind the two classes of bundles.  (In short, this story relates to the exact sequence 1 \rightarrow SO(n) \rightarrow O(n) \stackrel{det}{\rightarrow} O(1) = \mathbb{Z}_2 \rightarrow 1; in just the same way we have big groups SO, Spin, and so forth.)

So spin structures have a story much like the above, but where the exact sequence 1 \rightarrow \mathbb{Z}_2 \rightarrow Spin(n) \rightarrow SO(n) \rightarrow 1 plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups.  A spin structure on some SO(n) bundle E, let’s say represented by f : X \rightarrow B(SO(n)) is thus, again, a lifting to \tilde{f} : X \rightarrow B(Spin(n)).  The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, w_2(E).  Hisham Sati also explained the example of “generalized” spin structures in these terms.  But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds.  These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.

The title’s term “higher spin structures” comes from the fact that we’ve so far had a tower of classifying spaces (or groups), B(O) \leftarrow B(SO) \leftarrow B(Spin), and so on.  Then the problem of putting various sorts of structures on X has been turned into the problem of lifting a map f : X \rightarrow S(O) up this tower.  At each point, the obstruction to lifting is some cohomology class with coefficients in the groups (O, SO, etc.)  So when are these structures interesting?

This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension.  In this case, this has to do with the homotopy groups  – of O and its cousins.  So it turns out that the homotopy group \pi_k(O) (which is the same as \pi_k(O_n) as long as n is bigger than k) follows a pattern, where \pi_k(O) = \mathbb{Z}_2 if k = 0,1 (mod 8), and \pi_k(O) = \mathbb{Z} if k = 3,7 (mod 8).  The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem.  These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into O from such a sphere.

All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions.  In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on X which may or may not exist.  Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions.  In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures.  The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes.  These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions.  It turns out that the \pi_k homotopy group is related to parallel transport along (k-1)-dimensional surfaces in X, which can be thought of as the world-sheets of (k-2)-dimensional “particles” (or rather, “branes”).

So, for instance, the fact that \pi_1(O) is nontrivial means there’s an obstruction to a lifting in the form of a class in H^2(X,\mathbb{Z}), which has to do with spin structure – as above.  “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for X must be zero.  The fact that \pi_3(O) = \mathbb{Z} is nontrivial means there’s an obstruction to a lifting in the form of a class in H^4(X, \mathbb{Z}).  This has to do with “string bundles”, where the string group is a higher analog of Spin in exactly the sense we’ve just described.  If such a lifting exists, then there’s a “string-structure” on X which is compatible with the spin structure we lifted (and with the orientation a level below that).  Similarly, \pi_7(O) = \mathbb{Z} being nontrivial, by way of an obstruction in H^8, means there’s an interesting notion of “five-brane” structure, and a Fivebrane group, and so on.  Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.

A slight refinement of the above, and actually more directly related to “cancellation” of the anomalies, is that these structures can be defined in a “twisted” way: given a cocycle in the appropriate cohomology group, we can ask that a lifting exist, not on the nose, but as a diagram commuting only up to a higher cell, which is exactly given by the cocycle.  I mentioned, in the previous section, a situation where the cocycle gives an associator, so that instead of being exactly associative, a structure has a “twisted” associativity.  This is similar, except we’re twisting the condition that makes a spin structure (or higher spin structure) well-defined.  So if X has the wrong characteristic class, we can only define one of these twisted structures at that level.

This theme of higher cohomology and gerbes, and their geometric interpretation, was another one that turned up throughout the talks in the workshop…

And speaking of that: coming up soon, some descriptions of the actual workshop.

I’d like to continue describing the talks that made up the HGTQGR workshop, in particular the ones that took place during the school portion of the event.  I’ll save one “school” session, by Laurent Freidel, to discuss with the talks because it seems to more nearly belong there. This leaves five people who gave between two and four lectures each over a period of a few days, all intermingled. Here’s a very rough summary in the order of first appearance:

2D Extended TQFT

Chris Schommer-Pries gave the longest series of talks, about the classification of 2D extended TQFT’s.  A TQFT is a kind of topological invariant for manifolds, which has a sort of “locality” property, in that you can decompose the manifold, compute the invariant on the parts, and find the whole by gluing the pieces back together.  This is expressed by saying it’s a monoidal functor Z : (Cob_d, \sqcup) \rightarrow (Vect, \otimes), where the “locality” property is now functoriality property that composition is preserved.  The key thing here is the cobordism category Cob_d, which has objects (d-1)-dimensional manifolds, and morphisms d-dimensional cobordisms (manifolds with boundary, where the objects are components of the boundary).  Then a closed d-manifold is just a cobordism from $latex\emptyset$ to itself.

Making this into a category is actually a bit nontrivial: gluing bits of smooth manifolds, for instance, won’t necessarily give something smooth.  There are various ways of handling this, such as giving the boundaries “collars”, but Chris’ preferred method is to give boundaries (and, ultimately, corners, etc.) a”halation”.  This word originally means the halo of light around bright things you sometimes see in photos, but in this context, a halation for X is an equivalence class of embeddings into neighborhoods U \subset \mathbb{R}^d.  The equivalence class says two such embeddings into U and V are equivalent if there’s a compatible refinement into some common W that embeds into both U and V.  The idea is that a halation is a kind of d-dimensional “halo”, or the “germ of a d-manifold” around X.  Then gluing compatibly along (d-1)-boundaries with halations ensures that we get smooth d-manifolds.  (One can also extend this setup so that everything in sight is oriented, or has some other such structure on it.)

In any case, an extended TQFT will then mean an n-functor Z : (Bord_d,\sqcup) \rightarrow (\mathcal{C},\otimes), where (\mathcal{C},\otimes) is some symmetric monoidal n-category (which is supposed to be similar to Vect).  Its exact nature is less important than that of Bord_d, which has:

  • 0-Morphisms (i.e. Objects): 0-manifolds (collections of points)
  • 1-Morphisms: 1-dimensional cobordisms between 0-manifolds (curves)
  • 2-Morphisms: 2-dim cobordisms with corners between 1-Morphisms (surfaces with boundary)
  • d-Morphisms: d-dimensional cobordisms between (d-1)-Morphisms (n-manifolds with corners), up to isomorphism

(Note: the distinction between “Bord” and “Cobord” is basically a matter of when a given terminology came in.  “Cobordism” and “Bordism”, unfortunately, mean the same thing, except that “bordism” has become popular more recently, since the “co” makes it sound like it’s the opposite category of something else.  This is kind of regrettable, but that’s what happened.  Sorry.)

The crucial point, is that Chris wanted to classify all such things, and his approach to this is to give a presentation of Bord_d.  This is based on stuff in his thesis.  The basic idea is to use Morse theory, and its higher-dimensional generalization, Cerf theory.  The idea is that one can put a Morse function  on a cobordism (essentially, a well-behaved “time order” on points) and look at its critical points.  Classifying these tells us what the generators for the category of cobordisms must be: there need to be enough to capture all the most general sorts of critical points.

Cerf theory does something similar, but one dimension up: now we’re talking about “stratified” families of Morse functions.  Again one studies critical points, but, for instance, on a 2-dim surface, there can be 1- and 0-dimensional parts of the set of cricical points.  In general, this gets into the theory of higher-dimensional singularities, catastrophe theory, and so on.  Each extra dimension one adds means looking at how the sets of critical points in the previous dimension can change over “time” (i.e. within some stratified family of Cerf functions).  Where these changes themselves go through critical points, one needs new generators for the various j-morphisms of the cobordism category.  (See some examples of such “catastrophes”, such as folds, cusps, swallowtails, etc. linked from here, say.)  Showing what such singularities can be like in the “generic” situation, and indeed, even defining “generic” in a way that makes sense in any dimension, required some discussion of jet bundles.  These are generalizations of tangent bundles that capture higher derivatives the way tangent bundles capture first-derivatives.  The essential point is that one can find a way to decompose these into a direct sum of parts of various dimensions (capturing where various higher derivatives are zero, say), and these will eventually tell us the dimension of a set of critical points for a Cerf function.

Now, this gives a characterization of what cobordisms can be like – part of the work in the theorem is to show that this is sufficient: that is, given a diagram showing the critical points for some Morse/Cerf function, one needs to be able to find the appropriate generators and piece together the cobordism (possibly a closed manifold) that it came from.  Chris showed how this works – a slightly finicky process involving cutting a diagram of the singular points (with some extra labelling information) into parts, and using a graphical calculus to work out how pasting works – and showed an example reconstruction of a surface this way.  This amounts to a construction of an equivalence between an “abstract” cobordism category given in terms of generators (and relations) which come from Cerf theory, and the concrete one.  The theorem then says that there’s a correspondence between equivalence classes of 2D cobordisms, and certain planar diagrams, up to some local moves.  To show this properly required a digression through some theory of symmetric monoidal bicategories, and what the right notion of equivalence for them is.

This all done, the point is that Bord_d has a characterization in terms of a universal property, and so any ETQFT Z : Bord_d \rightarrow \mathcal{C} amounts to a certain kind of object in \mathcal{C} (corresponding to the image of the point – the generating object in Bord_d).  For instance, in the oriented situation this object needs to be “fully dualizable”: it should have a dual (the point with opposite orientation), and a whole bunch of maps that specify the duality: a cobordism from (+,-) to nothing (just the “U”-shaped curve), which has a dual – and some 2-D cobordisms which specify that duality, and so on.  Specifying all this dualizability structure amounts to giving the image of all the generators of cobordisms, and determines the functors Z, and vice versa.

This is a rapid summary of six hours of lectures, of course, so for more precise versions of these statements, you may want to look into Chris’ thesis as linked above.

Homotopy QFT and the Crossed Menagerie

The next series of lectures in the school was Tim Porter’s, about relations between Homotopy Quantum Field Theory (HQFT) and various sort of crossed gizmos.  HQFT is an idea introduced by Vladimir Turaev, (see his paper with Tim here, for an intro, though Turaev also now has a book on the subject).  It’s intended to deal with similar sorts of structures to TQFT, but with various sorts of extra structure.  This structure is related to the “Crossed Menagerie”, on which Tim has written an almost unbelievably extensive bunch of lecture notes, of which a special short version was made for this lecture series that’s a mere 350 pages long.

Anyway, the cobordism category Bord_d described above is replaced by one Tim called HCobord(d,B) (see above comment about “bord” and “cobord”, which mean the same thing).  Again, this has d-dimensional cobordisms as its morphisms and (d-1)-dimensional manifolds as its objects, but now everything in sight is equipped with a map into a space B – almost.  So an object is X \rightarrow B, and a morphism is a cobordism with a homotopy class of maps M \rightarrow B which are compatible with the ones at the boundaries.  Then just as a d-TQFT is a representation (i.e. a functor) of Cob_d into Vect, a (d,B)-HQFT is a representation of HCobord(d,B).

The motivating example here is when B = B(G), the classifying space of a group.  These spaces are fairly complicated when you describe them as built from gluing cells (in homotopy theory, one typically things of spaces as something like CW-complexes: a bunch of cells in various dimensions glued together with face maps etc.), but B(G) has the property that its fundamental group is G, and all other homotopy groups are trivial (ensuring this part is what makes the cellular decomposition description tricky).

The upshot is that there’s a correspondence between (homotopy classes of) maps Map(X ,B(G)) \simeq Hom(\pi(X),G) (this makes a good alternative definition of the classifying space, though one needs to ).  Since a map from the fundamental group into G amounts to a flat principal G-bundle, we can say that HCobord(d,B(G)) is a category of manifolds and cobordisms carrying such a bundle.  This gets us into gauge theory.

But we can go beyond and into higher gauge theory (and other sorts of structures) by picking other sorts of B.  To begin with, notice that the correspondence above implies that mapping into B(G) means that when we take maps up to homotopy, we can only detect the fundamental group of X, and not any higher homotopy groups.  We say we can only detect the “homotopy 1-type” of the space.  The “homotopy n-type” of a given space X is just the first n homotopy groups (\pi_1(X), \dots, \pi_n(X)).  Alternatively, an “n-type” is an equivalence class of spaces which all have the same such groups.  Or, again, an “n-type” is a particular representative of one of these classes where these are the only nonzero homotopy groups.

The point being that if we’re considering maps X \rightarrow B up to homotopy, we may only be detecting the n-type of X (and therefore may as well assume X is an n-type in the last sense when it’s convenient).  More precisely, there are “Postnikov functors” P_n(-) which take a space X and return the corresponding n-type.  This can be done by gluing in “patches” of higher dimensions to “fill in the holes” which are measured by the higher homotopy groups (in general, the result is infinite dimensional as a cell complex).  Thus, there are embeddings X \hookrightarrow P_n(X), which get along with the obvious chain

\dots \rightarrow P_{n+1}(X) \rightarrow P_n(X) \rightarrow P_{n-1}(X) \rightarrow \dots

There was a fairly nifty digression here explaining how this is a “coskeleton” of X, in that P_n is a right adjoint to the “n-skeleton” functor (which throws away cells above dimension n, not homotopy groups), so that S(Sk_n(M),X) \cong S(M,P_n(X)).  To really explain it properly, though I would have to really explain what that S is (it refers to maps in the category of simplicial sets, which are another nice model of spaces up to homotopy).  This digression would carry us away from higher gauge theory, which is where I’m going.

One thing to say is that if X is d-dimensional, then any HQFT is determined entirely by the d-type of B.  Any extra jazz going on in B‘s higher homotopy groups won’t be detected when we’re only mapping a d-dimensional space X into it.  So one might as well assume that B is just a d-type.

We want to say we can detect a homotopy n-type of a space if, for example, B = B(\mathcal{G}) where \mathcal{G} is an “n-group”.  A handy way to account for this is in terms of a “crossed complex”.  The first nontrivial example of this would be a crossed module, which consists of

  • Two groups, G and H with
  • A map \partial : H \rightarrow G and
  • An action of G on H by automorphisms, G \rhd H
  • all such that action looks as much like conjugation as possible:
    • \partial(g \rhd h) = g (\partial h) g^{-1} (so that \partial is G-equivariant)
    • \partial h \rhd h' = h h' h^{-1} (the “Peiffer identity”)

This definition looks a little funny, but it does characterize “2-groups” in the sense of categories internal to \mathbf{Groups} (the definition used elsewhere), by taking G to be the group of objects, and H the group of automorphisms of the identity of G.  In the description of John Huerta’s lectures, I’ll get back to how that works.

The immediate point is that there are a bunch of natural examples of crossed modules.  For instance: from normal subgroups, where \partial: H \subset G is inclusion and the action really is conjugation; from fibrations, using fundamental groups of base and fibre; from a canonical case where H = Aut(G)  and \partial = 1 takes everything to the identity; from modules, taking H to be a G-module as an abelian group and \partial = 1 again.  The first and last give the classical intuition of these guys: crossed modules are simultaneous generalizations of (a) normal subgroups of G, and (b) G-modules.

There are various other examples, but the relevant thing here is a theorem of MacLane and Whitehead, that crossed modules model all connected homotopy 2-types.  That is, there’s a correspondence between crossed modules up to isomorphism and 2-types.  Of course, groups model 1-types: any group is the fundmental group for a 1-type, and any 1-type is the classifying space for some group.  Likewise, any crossed module determines a 2-type, and vice versa.  So this theorem suggests why crossed modules might deserve to be called “2-groups” even if they didn’t naturally line up with the alternative definition.

To go up to 3-types and 4-types, the intuitive idea is: “do for crossed modules what we did for groups”.  That is, instead of a map of groups \partial : H \rightarrow G, we consider a map of crossed modules (which is given by a pair of maps between the groups in each) and so forth.  The resulting structure is a square diagram in \mathbf{Groups} with a bunch of actions.  Each of these maps is the \partial map for a crossed module.  (We can think of the normal subgroup situation: there are two normal subgroups H,K of G, and in each of them, the intersection H \cap K is normal, so it determines a crossed module).  This is a “crossed square”, and things like this correspond exactly to homotopy 3-types.  This works roughly as before, since there is a notion of a classifying space B(\mathcal{G}) where \mathcal{G} =   (G,H,\partial,\rhd), and similarly on for crossed n-cubes.   We can carry on in this way to define a “crossed n-cube”, which correspond to homotopy (n+1)-types.  The correspondence is a little bit more fiddly than it was for groups, but it still exists: any (n+1)-type is the classifying space for a crossed n-cube, and any such crossed n-cube has an (n+1)-type for its classifying space.

This correspondence is the point here.  As we said, when looking at HQFT’s from HCobord(d,B), we may as well assume that B is a d-type.  But then, it’s a classifying space for some crossed (d-1)-cube.  This is a sensible sort of B to use in an HQFT, and it ends up giving us a theory which is related to higher gauge theory: a map X \rightarrow B(\mathcal{G}) up to homotopy, where \mathcal{G} is a crossed n-cube will correspond to the structure of a flat (n+1)-bundle on X, and similarly for cobordisms.  HQFT’s let us look at the structure of this structured cobordism category by means of its linear representations.  Now, it may be that this crossed-cube point of view isn’t the best way to look at B, but it is there, and available.

To say more about this, I’ll have to talk more directly about higher gauge theory in its own terms – which I’ll do in part IIb, since this is already pretty long.

A more substantial post is upcoming, but I wanted to get out this announcement for a conference I’m helping to organise, along with Roger Picken, João Faria Martins, and Aleksandr Mikovic.  Its website: https://sites.google.com/site/hgtqgr/home has more details, and will have more as we finalise them, but here are some of them:

Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity

Lisbon, 10-13 February, 2011 (Workshop), 7-13 February, 2011 (School)

Description from the website:

Higher gauge theory is a fascinating generalization of ordinary abelian and non-abelian gauge theory, involving (at the first level) connection 2-forms, curvature 3-forms and parallel transport along surfaces. This ladder can be continued to connection forms of higher degree and transport along extended objects of the corresponding dimension. On the mathematical side, higher gauge theory is closely tied to higher algebraic structures, such as 2-categories, 2-groups etc., and higher geometrical structures, known as gerbes or n-gerbes with connection. Thus higher gauge theory is an example of the categorification phenomenon which has been very influential in mathematics recently.

There have been a number of suggestions that higher gauge theory could be related to (4D) quantum gravity, e.g. by Baez-Huerta (in the QG^2 Corfu school lectures), and Baez-Baratin-Freidel-Wise in the context of state-sums. A pivotal role is played by TQFTs in these approaches, in particular BF theories and variants thereof, as well as extended TQFTs, constructed from suitable geometric or algebraic data. Another route between higher gauge theory and quantum gravity is via string theory, where higher gauge theory provides a setting for n-form fields, worldsheets for strings and branes, and higher spin structures (i.e. string structures and generalizations, as studied e.g. by Sati-Schreiber-Stasheff). Moving away from point particles to higher-dimensional extended objects is a feature both of loop quantum gravity and string theory, so higher gauge theory should play an important role in both approaches, and may allow us to probe a deeper level of symmetry, going beyond normal gauge symmetry.

Thus the moment seems ripe to bring together a group of researchers who could shed some light on these issues. Apart from the courses and lectures given by the invited speakers, we plan to incorporate discussion sessions in the afternoon throughout the week, for students to ask questions and to stimulate dialogue between participants from different backgrounds.

Provisional list of speakers:

  • Paolo Aschieri (Alessandria)
  • Benjamin Bahr (Cambridge)
  • Aristide Baratin (Paris-Orsay)
  • John Barrett (Nottingham)
  • Rafael Diaz (Bogotá)
  • Bianca Dittrich (Potsdam)
  • Laurent Freidel (Perimeter)
  • John Huerta (California)
  • Branislav Jurco (Prague)
  • Thomas Krajewski (Marseille)
  • Tim Porter (Bangor)
  • Hisham Sati (Maryland)
  • Christopher Schommer-Pries (MIT)
  • Urs Schreiber (Utrecht)
  • Jamie Vicary (Oxford)
  • Konrad Waldorf (Regensburg)
  • Derek Wise (Erlangen)
  • Christoph Wockel (Hamburg)

The workshop portion will have talks by the speakers above (those who can make it), and any contributed talks.  The “school” portion is, roughly, aimed at graduate students in a field related to the topics, but not necessarily directly in them.  You don’t need to be a student to attend the school, of course, but they are the target audience.  The only course that has been officially announced so far will be given by Christopher Schommer-Pries, on TQFT.  We hope/expect to also have minicourses on Higher Gauge Theory, and Quantum Gravity as well, but details aren’t settled yet.

If you’re interested, the deadline to register is Jan 8 (hence the rush to announce).  Some funding is available for those who need it.

In the most recent TQFT Club seminar, we had a couple of talks – one was the second in a series of three by Marco Mackaay, which as promised previously I’ll write up together after the third one.

The other was by Björn Gohla, a student of João Faria Martins, giving an overview on the subject of “Tricategories and Trifunctors”, a mostly expository talk explaining some definitions.  Actually, this was a bit more specific than a general introduction – the point of it was to describe a certain kind of mapping space.  I’ve talked here before about representing the “configuration space” of a gauge theory as a groupoid: the objects are (optionally, flat) connections on a manifold M, and the morphisms are gauge transformations taking one connection to another.  The reason for the things Björn was talking about is analogous, except that in this case, the goal is to describe the configuration space of a higher gauge theory.

There are at least two ways I know of to talk about higher gauge theory.  One is in terms of categorical (or n-categorical) groups – which makes it a “categorification” of gauge theory in the sense of reproducing in \mathbf{Cat} (or \mathbf{nCat}) an analog of a sturcture, gauge theory, originally formulated in \mathbf{Set}.  Among other outlines, you might look at this one by John Baez and John Huerta for an introduction.  Another uses the lingo of crossed modules or crossed complexes.  In either case, the essential point is the same: there is some collection of groups (or groupoids, but let’s say groups to keep everything clear) which play the role of the single gauge group in ordinary gauge theory.

In the first language, we can speak of a “2-group”, or “categorical group” – a group internal to \mathbf{Cat}, or what is equivalent, a category internal to \mathbf{Grp}, which would have a group of objects and a group of morphisms (and, in higher settings still, groups of 2-morphisms, 3-morphisms, and so on).  The structure maps of the category (source, target, composition, etc.) have to live in the category of groups.

A crossed complex of groups (again, we could generalize to groupoids, but I won’t) is a nonabelian variation on a chain complex: a sequence of groups with maps from one to the next.  There are also a bunch more structures, which ultimately serve to reproduce all the kind of composition, source, and target maps in the n-categorical groups: some groups act on others, there are “bracket” operations on one group valued in another, and so forth.  This paper by Brown and Higgins explains how the two concepts are related when most of the groups are abelian, and there’s a lot more about crossed complexes and related stuff in Tim Porter’s “Crossed Menagerie“.

The point of all this right now is that these things play the role of the gauge group in higher gauge theory.  The idea is that in gauge theory, you have a connection.  Typically this is described in terms of a form valued in the Lie algebra of the gauge group.  Then a (thin) homotopy classes of curves, gets a holonomy valued in the group by integrating that form.  Alternatively, you can just think of the path groupoid of a manifold \mathcal{P}_1(M), where those classes of curves form the morphisms between the objects, which are just points of M.  Then a connection defines a functor \Gamma : \mathcal{P}_1(M) \rightarrow G, where G is the gauge group thought of as a category (groupoid in fact) with one object.  Or, you can just define a connection that way in the first place.  In higher gauge theory, a similar principle exists: begin with the n-path groupoid \mathcal{P}_n(M) where the morphisms are (thin homotopy classes of) paths, the 2-morphisms are surfaces (really homotopy classes of homotopies of paths), and so on, so the k-morphisms are k-dimensional bits of M.  Then you could define an n-connection as a n-functor into an n-group as defined above.  OR, you could define it in terms of a tower of differential k-forms valued in the crossed complex of Lie algebras associated to the crossed complex of Lie groups that replaces the gauge group.  You can then use an integral to get an element of the group at level k of the complex for any given k-morphism in \mathcal{P}_n(M), which (via the equivalence I mentioned) amounts to the same thing as the other definition of connection.

João Martins has done some work on this sort of thing when n is dimension 2 (with Tim Porter) and 3 (with Roger Picken), which I guess is how Björn came to work on this question.  The question is, roughly, how to describe the moduli space of these connections.  The gist of the answer is that it’s a functor n-category [\mathcal{P}_n(M),\mathcal{G}], where \mathcal{G} is the n-group.  A little more generally, the question is how to describe mapping spaces for higher categories.  In particular, he was talking about the case n=3, which is where certain tricky issues start to show up.  In particular every bicategory (the weakest form of 2-category) is (bi)equivalent to a strict 2-category, so there’s no real need to worry about weakening things like associativity so that they only work up to isomorphism – these are all equalities.  With 3-categories, this fails: the weakest kind of 3-category is a tricategory (introduced by Gordon, Power and Street, though also see the references beyond that link).  These are always tri-equivalent to something stricter than general, but not completely strict: Gray-categories.  The only equation from 2-categories which has to be weakened to an isomorphism here is the interchange law: given a square of four morphisms, we can either compose vertically first, and then horizontally, or vice versa.  In a Gray-category, there’s an “interchanger” isomorphism

I_{\alpha,\alpha ',\beta,\beta'} : (\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') \Rightarrow (\alpha \cdot \alpha ') \circ (\beta \cdot \beta ')

where \cdot is vertical composition of 2-cells, and \circ is horizontal (i.e. the same direction as 1-cells).  This is supposed to satisfy a compatibility condition.  It’s essentially the only one you can come up with starting with (\alpha \cdot \alpha ') \circ \beta (and composing it in different orders by throwing in identities in various places).

There’s another way to look at things, as Björn explained, in terms of enriched category theory.  If you have a monoidal category (\mathcal{V},\otimes), then a (\mathcal{V},\otimes)-enriched category \mathbb{G} is one in which, for any two objects x,y, there is an object \mathbb{G}(x,y) \in \mathcal{V} of morphisms, and composition gives morphisms \circ_{x,y,z} : \mathbb{G}(y,z) \otimes \mathbb{G}(x,y) \rightarrow \mathbb{G}(x,z).  A strict 3-category is enriched in \mathbf{2Cat}, with its usual tensor product, dual to its internal hom [-,-] (which gives the mapping 2-category of functors, natural transformations, and modifications, between any two 2-categories).  A Gray category is similar, except that it is enriched in \mathbf{Gray}, a version of \mathbf{2Cat} with a different tensor product, dual to the hom functor [-,-]' which gives the mapping 2-category with pseudonatural transformations (the weak version of the concept, where the naturality square only has to commute up to a specified 2-cell) as morphisms.  These are not the same, which is where the unavoidability of weakening 3-categories “really” comes from.   The upshot of this is as above: it matters which order we compose things in.

Having defined Gray-categories, let’s say A and B (which, in the applications I mentioned above, tend to actually be Gray-groupoids, though this doesn’t change the theory substantially), the point is to talk about “mapping spaces” – that is, Gray-categories of Gray-functors (etc.) from A to B.

Since they’ve been defined terms of enriched category theory, one wants to use the general theory of enriched functors, transformations, and so forth – which is a lot easier than trying to work out the correct definitions from scratch using a low-level description.  So then a Gray-functor F : A \rightarrow B has an object map F_0 : A_0 \rightarrow B_0, mapping objects of A to objects of B, and then for each x,y \in A_0, a morphism in \mathbf{Gray} (which is our \mathcal{V}), namely F_{x,y} : A(x,y) \rightarrow B(F(x),F(y)).  There are a bunch of compatibility conditions, which can be expressed for any monoidal category \mathcal{V} (since they involve diagrams with the map \circ_{x,y,z} for any triple, and the like).  Similar comments apply to defining \mathcal{V}-natural transformations.

There is a slight problem here, which is that in this case, \mathcal{V} = \mathbf{Gray} is a 2-category, so we really need to use a form of weakly enriched categories…  All the compatibility diagrams should have 2-cells in them, and so forth.  This, too, gets complicated.  So Björn explained is a shortcut from drawing n-dimensional diagrams for these mapping n-categories in terms of the arrow category \vec{B}. This is the category whose objects are the morphisms of B, and whose morphisms are commuting squares, or when B is a 2-category, squares with a 2-cell, so a morphism in \vec{B} from f: x \rightarrow y to f' : x' \rightarrow y' is a triple g = (g_x,g_y,g_f) like so:

Morphism in arrow category

The 2-morphisms in \vec{B} are commuting “pillows”, where the front and back face are morphisms like the above. So \beta : g \Rightarrow g' is \beta = (\beta_x,\beta_y), where \beta_x : g_x \Rightarrow g_{x'} is a 2-cell, and the whole “pillow” commutes.  When B is a tricategory, then we need to go further – these 2-morphsims should be triples including a 3-cell \beta_f filling the “pillow”, and then 3-morphisms are commuting structures between these. These diagrams get hard to draw pretty quickly. This is the point of having an ordinary 2D diagram with at most 1-dimensional cells: pushing all the nasty diagrams into these arrow categories, we can replace a 2-cell representing a natural transformation with a diagram involving the arrow category.

This uses that there are source and target maps (which are Gray-functors, of course) which we’ll call d_0, d_1: \vec{B} \rightarrow B. So then here (in one diagram) we have two ways of depicting a natural transformation \alpha :  F \rightarrow G between functors F,G : A \Rightarrow B:

One is the 2-cell, and the other is the functor into \vec{B}, such that d_0 \circ \alpha = F and d_1 \circ \alpha = G.
To depict a modification between natural transformations (a 3-cell between 2-cells) just involves building the arrow category of \vec{B}, say \vec{\vec{B}}, and drawing an arrow from A into it. And so on: in principle, there is a tower above B built by iterating the arrow category construction, and all the different levels of “functor”, “natural transformation”, “modification”, and all the higher equivalents are just functors into different levels of this tower.  (The generic term for the k^{th} level of maps-between-maps-etc between n-categories is “(n,k)-transfor“, a handy term coined here.)
The advantage here is that at least the general idea can be extended pretty readily to higher values of n than 3.  Naturally, no matter which way one decides to do it, things will get complicated – either there’s a combinatorial explosion of things to consider, or one has to draw higher-dimensional diagrams, or whatever.  This exploding complexity of n-categories (in this case, globular ones) is one of the reasons why simplicial appreaches – quasicategories or \infty-categories – are good.  They allow you to avoid talking about those problems, or at least fold them into fairly well-understood aspects of simplicial sets.  A lot of things – limits, colimits, mapping spaces, etc. are pretty well understood in that case (see, for instance, the first chapter of Joshua Nicholls-Barrer’s thesis for the basics, or Jacob Lurie’s humongous book for something more comprehensive).  But sometimes, as in this case, they just don’t happen to be the things you want for your application.  So here we have some tools for talking about mapping spaces in the world of globular n-categories – and as the work by Martins/Porter/Picken show, it’s motivated by some fairly specific work about invariants of manifolds, differential geometry, and so on.

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