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.