### conferences

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.

So I had a busy week from Feb 7-13, which was when the workshop Higher Gauge Theory, TQFT, and Quantum Gravity (or HGTQGR) was held here in Lisbon.  It ended up being a full day from 0930h to 1900h pretty much every day, except the last.  We’d tried to arrange it so that there were coffee breaks and discussion periods, but there was also a plethora of talks.  Most of the people there seemed to feel that it ended up pretty well.  Since then I’ve been occupied with other things – family visiting the country, for one, so it’s taken a while to get around to writing it up.  Since there were several parts to the event, I’ll do this in several parts as well, of which this is the first one.

Part of the point of the workshop was to bring together a few related subjects in which category theoretic ideas come into areas of mathematics which play a role in physics, and hopefully to build some bridges toward applications.  While it leaned pretty strongly on the mathematical side of this bridge, I think we did manage to get some interaction at the overlap.  Roger Picken drew a nifty picture on the whiteboard at the end of the workshop summarizing how a lot of the themes of the talks clustered around the three areas mentioned in the title, and suggesting how TQFT really does form something of a bridge between the other two – one reason it’s become a topic of some interest recently.  I’ll try to build this up to a similar punchline.

### Pre-School

Before the actual event began, though, we had a bunch of talks at IST for a local audience, to try to explain to mathematicians what the physics part of the workshop was about.  Aleksandr Mikovic gave a two-talk introduction to Quantum Gravity, and Sebastian Guttenberg gave a two-part intro to String Theory.  These are two areas where higher gauge theory (in the form of n-connections and n-bundles, or of n-gerbes) has made an appearance, and were the main physics content of the workshop talks.  They set up the basics to help put those talks in context.

Quantum Gravity

Aleksandr’s first talk set out the basic problem of quantizing the gravitational field (this isn’t the only attitude to what the problem of quantum gravity is, but it’s a good starting point), starting with the basic ingredients.  He summarized how general relativity describes gravity in terms of a metric $g_{\mu \nu}$ which is supposed to satisfy the Einstein equation, relating the curvature of the metric to a source field $T_{\mu \nu}$ which comes from matter.  Quantization then, starting from a classical picture involving trajectories of particles (or sections of fibre bundles to describe fields), one gets a picture where states are vectors in a Hilbert space, and there’s an algebra of operators including observables (self-adjoint operators) and time-evolution (hermitian ones).   An initial try at quantum gravity was to do this using the metric as the field, using the methods of perturbative QFT: treating the metric in terms of “small” fluctuations from some background metric like the flat Minkowski metric.  This uses the Einstein-Hilbert action $S=\frac{1}{G} \int \sqrt{det(g)}R$, where $G$ is the gravitational constant and $R$ is the Ricci scalar that summarizes the curvature of $g$.  This runs into problems: things diverge in various calculations, and since the coupling constant $G$ has units, one can’t “renormalize” the divergences away.  So one needs a non-perturbative approach,  one of which is “canonical quantization“.

After some choice of coordinates (so-called “lapse” and “shift” functions), this involves describing the action in terms of the (space part of) the metric $g_{kl}$ and some canonically conjugate “momentum” variables $\pi_{kl}$ which describe its extrinsic curvature.  The Euler-Lagrange equations (found as usual by variational calculus methods) then turn out to give the “Hamiltonian constraint” that certain functions of $g$ are always zero.  Then the program is to get a Poisson algebra giving commutators of the $\pi$ and $g$ variables, then turn it into an algebra of operators in a standard way.  This also runs into problems because the space of metrics isn’t a Hilbert space.  One solution is to not use the metric, but instead a connection and a “frame field” – the so-called Ashtekar variables for GR.  This works better, and gives the “Loop Quantum Gravity” setup, since observables tend to be expressed as holonomies around loops.

Finally, Aleksandr outlined the spin foam approach to quantizing gravity.  This is based on the idea of a quantum geometry as a network (graph) with edges labelled by spins, i.e. representations of SU(2) (which are labelled by half-integers).  Vertices labelled by intertwining operators (which imposes triangle inequalities, as it happens).  The spin foam approach takes a Hilbert space with a basis given by these spin networks.  These are supposed to be an alternative way of describing geometries given by SU(2)-connections. The representations arise because, as the Peter-Weyl theorem shows, they form a nice basis for $L^2(SU(2))$.  Then to get operators associated to “foams” that interpolate the spacetime between two such geometries (i.e. linear combinations of spin networks).  These are 2-complexes where faces are labelled with spins, and edges with intertwiners for the spins on the faces incident to them.  The operators arise from  a discrete variant of the Feynman path-integral, where time-evolution comes from integrating an action over a space of (classical) trajectories, which in this case are foams.  This needs an action to integrate – in the discrete world, this corresponds to ways of choosing weights $A_e$ for edges and $A_f$ for faces in a generic partition function:

$Z = \sum_{J,I} \prod_{faces} A_f(j_f) \prod_{edges}A_e(i_l)$

which is a sum over the labels for representations and intertwiners.  Some of the talks that came later in the conference (e.g. by Benjamin Bahr and Bianca Dittrich) came back to discuss principles behind how these $A$ functions could be chosen.  (Aristide Baratin’s talk described a similar but more general kind of model based on 2-groups.)

String Theory

In parallel with these, Sebastian Guttenberg gave us a two-lecture introduction to string theory.  His starting point is the intuition that a lot of classical physics studies particles living on a background of some field.  The field can be understood as an approximate way of talking about a large number of quantum-mechanical particles, rather as the dynamics of a large number of classical particles can be approximated by the equations of state for a fluid or gas (depending on how much they interact with one another, among other things).  In string theory and “string field theory”, we have a similar setup, except we replace the particles with small strings – either open strings (which look like intervals) or closed ones (which look like circles).

To begin with, he introduced the basic tools of “classical” string theory – the analog of classical mechanics of point particles.  This is the string analog of the following: one can describe a moving particle by its worldline – a path $x : \mathbb{R} \rightarrow M^{(D)}$ from a “generic” worldline into a ($D$-dimensional) manifold $M^{(D)}$.  This $M^{(D)}$ is generally taken to be like physical spacetime, which in this context means that it has a metric $g$ with signature $(-1,1,\dots,1)$ (that is, locally there’s a basis for tangent spaces with one timelike vector and $D-1$ spacelike ones).  Then one can define an action for a moving particle which is just determined by the length of the line’s image.  The nicest way to say this is $S[x] = m \int d\tau \sqrt{x*g}$, where $x*g$ means the pullback of the metric along the map $x$, $\tau$ is some parameter along the generic worldline, and $m$, the particle’s mass, is a coupling constant which doesn’t happen to affect the result in this simple case, but eventually becomes important.  One can do the usual variational-calculus of the Lagrangian approach here, finding a critical point of the action occurs when the particle is travelling in a geodesic – a straight line, in flat space, or the closest available approximation.  In paritcular, the Euler-Lagrange equations say that the covariant derivative of the path should be zero.

There’s an analogous action for a string, the Nambu-Goto action.  Instead of a single-parameter $x$, we now have an embedding of a “generic string worldsheet” – let’s say $\Sigma^{(2)} \cong S^1 \times \mathbb{R}$ into spacetime: $x : \Sigma^{(2)} \rightarrow M^{(D)}$.  Then then the analogous action is just $S[x] = \int_{\Sigma^{(2)}} \star_{x*g} 1$.  This is pretty much the same as before: we pull back the metric to get $x*g$, and integrate over the generic worldsheet.  A slight subtlety comes because we’re taking the Hodge dual $\star$.  This is conceptually clean, but expands out to a fairly big integral when you express it in coordinates, where the leading term  involves $\sqrt{det(\partial_{\mu} x^m \partial_{\nu} x^n g_{mn}}$ (the determinant is taken over $(\mu,\nu)$.  Varying this to get the equations of motion produces:

$0 = \partial_{\mu} \partial^{\mu} x^k + \partial_{\mu} x^m \partial^{\mu} x^n \Gamma_{mn}^k$

which is the two-dimensional analog of the geodesic equation for a point particle (the $\Gamma$ are the Christoffel symbols associated to the connection that goes with the metric).  The two-dimensional analog says we have a critical point for the area of the surface which is the image of $\Sigma^{(2)}$ – in fact, a “maximum”, given the sign of the metric.  For solutions like this, the pullback metric on the worldsheet, $x*g$, looks flat.  (Naturally, the metric looks flat along a geodesic, too, but this is stronger in 2 dimensions, where there can be intrinsic curvature.)

A souped up version of the Nambu-Goto action is the Polyakov action, which is a natural variation that comes up when $\Sigma^{(2)}$ has a metric of its own, $h$.  You can check out the details behind that link, but part of what makes this action nice is that the corresponding Euler-Lagrange equation from varying $h$ says that $x*g \sim h$.  That is, the worldsheet $\Sigma^{(2)}$ will have an image with a shape such that its own metric agrees with the one induced from the spacetime $M^{(D)}$.   This action is called the Polyakov action (even though it was introduced by Deser and Zumino, among others) because Polyakov used it for quantizing the string.

Other variations on this action add additional terms which represent fields which the string might be affected by: a scalar $\phi(x)$, and a 2-form field $B_{mn}(x)$ (here we’re using the physics convention where $x$ represents both the function, and its values at particular points, in this case, values of parameters $(\sigma_0,\sigma_1)$ on $\Sigma^{(2)}$).

That 2-form, the “B-field”, is an important field in string theory, and eventually links up with higher gauge theory, which we’ll get to as we go on: one can interpret the B-field as part of a higher connection, to which the string is coupled (as in Baez and Perez, say).  The scalar field $\phi$ essentially determines how strongly the shape of the string itself affects the action – it’s a “string coupling” term, or string coupling “constant” if it’s chosen to be just a number $\phi_0$.  (In such a case, the action includes a term that looks like $\phi_0$ times the Euler characteristic of the surface $\Sigma^{(2)}$.)

Sebastian briefly explained some of the physical intuition for why these are the kinds of couplings which it makes sense to introduce.  Essentially, any coupling one writes in coordinates has to get along with gauge symmetries, changes of coordinates, etc.  That is, there should be no physical difference between the class of solutions one finds in a given set of coordinates, and the coordinates one gets by doing some diffeomorphism on the spacetime $M^{(D)}$, or by changing the metric on $\Sigma^{(2)}$ by some conformal transformation $h_{\mu \nu} \mapsto exp(2 \omega(\sigma^0,\sigma^1)) h_{\mu \nu}$ (that is, scaling by some function of position on the worldsheet – underlying string theory is Conformal Field Theory in that the scale of the generic worldsheet is irrelevant – only the light-cones).  Anything a string couples to should be a field that transforms in a way that respects this.  One important upshot for the quantum theory is that when one quantizes a string coupled to such a field, this makes sure that time evolution is unitary.

How this is done is a bit more complicated than Sebastian wanted to go into in detail (and I got a little lost in the summary) so I won’t attempt to do it justice here.  The end results include a partition function:

$Z = \sum_{topologies} dx dh \frac{exp(-S[x,h])}{V_{diff} V_{weyl}}$

Remember: if one is finding amplitudes for various observables, the partition function is a normalizing factor, and finding the value of any observables means squeezing them into a similar-looking integral (and normalizing by this factor).  So this says that they’re found by summing over all the string topologies which go from the input to the output, and integrating over all embeddings $x : \Sigma^{(2)} \rightarrow M^{(D)}$ and metrics on $\Sigma^{(2)}$.  (The denominator in that fraction is dividing out by the volumes of the symmetry groups, as usual is quantum field theory since these symmetries mean one is “overcounting” physically identical situations.)

This is just the beginning of string field theory, of course: just as the dynamics of a free moving particle, or even a particle coupled to a background field, are only the beginning of quantum field theory.  But many later additions can be understood as adding various terms to the action $S$ in some such formalism.  These would be analogs of giving a point-particle attributes like charge, spin, “colour” and so forth in the Standard Model: these define how it couples to, hence is affected by, various kinds of fields.  Such fields can be understood in terms of connections (or, in general, higher connections, as we’ll get to later), which define how structures are “parallel-transported” along a path (or higher-dimensional surface).

Coming up in In Part II… I’ll summarize the School portion of the HGTQGR workshop, including lecture series by: Christopher Schommer-Pries on Classifying 2D Extended TQFT, which among other things explained Chris’ proof of the Cobordism Hypothesis using Cerf theory; Tim Porter on Homotopy QFT and the “Crossed Menagerie”, which describe a general framework for talking about quantum theories on cobordisms with structure; John Huerta on Higher Gauge Theory, which gave an introductory account of 2-groups and 2-bundles with 2-connections; Christoph Wockel on connections between Higher Gauge Theory and Infinite Dimensional Lie Theory, which described how some infinite-dimensional Lie algebras can’t be integrated to Lie groups, but only to 2-groups; and one by Hisham Sati on Higher Spin Structures in String Theory, which among other things described how cohomological obstructions to putting certain kinds of structure on manifolds motivates the use of particular higher dimensions.

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 first week of November, I was in Montreal for the biannual meeting of the Philosophy of Science Association, at the invitation of Hans Halvorson and Steve Awodey.  This was for a special session called “Category Theoretical Reflections on the Foundations of Physics”, which also had talks by Bob Coecke (from Oxford), Klaas Landsman (from Radboud University in Nijmegen), and Gonzalo Reyes (from the University of Montreal).  Slides from the talks in this session have been collected here by Steve Awodey.  The meeting was pretty big, and there were a lot of talks on a lot of different topics, some more technical, and some less.  There were enough sessions relating to physics that I had a full schedule just attending those, although for example there were sessions on biology and cognition which I might otherwise have been interested in sitting in on, with titles like “Biology: Evolution, Genomes and Biochemistry”, “Exploring the Complementarity between Economics and Recent Evolutionary Theory”, “Cognitive Sciences and Neuroscience”, and “Methodological Issues in Cognitive Neuroscience”.  And, of course, more fundamental philosophy of science topics like “Fictions and Scientific Realism” and “Kinds: Chemical, Biological and Social”, as well as socially-oriented ones such as “Philosophy of Commercialized Science” and “Improving Peer Review in the Sciences”.  However, interesting as these are, one can’t do everything.

In some ways, this was a really great confluence of interests for me – physics and category theory, as seen through a philosophical lens.  I don’t know exactly how this session came about, but Hans Halvorson is a philosopher of science who started out in physics (and has now, for example, learned enough category theory to teach the course in it offered at Princeton), and Steve Awodey is a philosopher of mathematics who is interested in category theory in its own right.  They managed to get this session brought in to present some of the various ideas about the overlap between category theory and physics to an audience mostly consisting of philosophers, which seems like a good idea.  It was also interesting for me to get a view into how philosophers approach these subjects – what kind of questions they ask, how they argue, and so on.  As with any well-developed subject, there’s a certain amount of jargon and received ideas that people can refer to – for example, I learned the word and current usage (though not the basic concept) of supervenience, which came up, oh, maybe 5-10 times each day.

There are now a reasonable number of people bringing categorical tools to bear on physics – especially quantum physics.  What people who think about the philosophy of science can bring to this research is the usual: careful, clear thinking about the fundamental concepts involved in a way that tries not to get distracted by the technicalities and keep the focus on what is important to the question at hand in a deep way.  In this case, the question at hand is physics.  Philosophy doesn’t always accomplish this, of course, and sometimes get sidetracked by what some might call “pseudoquestions” – the kind of questions that tend to arise when you use some folk-theory or simple intuitive understanding of some subtler concept that is much better expressed in mathematics.  This is why anyone who’s really interested in the philosophy of science needs to learn a lot about science in its own terms.  On the whole, this is what they actually do.

And, of course, both mathematicians and physicists try to do this kind of thinking themselves, but in those fields it’s easy – and important! – to spend a lot of time thinking about some technical question, or doing extensive computations, or working out the fiddly details of a proof, and so forth.  This is the real substance of the work in those fields – but sometimes the bigger “why” questions, that address what it means or how to interpret the results, get glossed over, or answered on the basis of some superficial analogy.  Mind you – one often can’t really assess how a line of research is working out until you’ve been doing the technical stuff for a while.  Then the problem is that people who do such thinking professionally – philosophers – are at a loss to understand the material because it’s recent and technical.  This is maybe why technical proficiency in science has tended to run ahead of real understanding – people still debate what quantum mechanics “means”, even though we can use it competently enough to build computers, nuclear reactors, interferometers, and so forth.

Anyway – as for the substance of the talks…  In our session, since every speaker was a mathematician in some form, they tended to be more technical.  You can check out the slides linked to above for more details, but basically, four views of how to draw on category theory to talk about physics were represented.  I’ve actually discussed each of them in previous posts, but in summary:

• Bob Coecke, on “Quantum Picturalism”, was addressing the monoidal dagger-category point of view, which looks at describing quantum mechanical operations (generally understood to be happening in a category of Hilbert spaces) purely in terms of the structure of that category, which one can see as a language for handling a particular kind of logic.  Monoidal categories, as Peter Selinger as painstakingly documented, can be described using various graphical calculi (essentially, certain categories whose morphisms are variously-decorated “strands”, considered invariant under various kinds of topological moves, are the free monoidal categories with various structures – so anything you can prove using these diagrams is automatically true for any example of such categories).  Selinger has also shown that, for the physically interesting case of dagger-compact closed monoidal categories, a theorem is true in general if and only if it’s true for (finite dimensional) Hilbert spaces, which may account for why Hilbert spaces play such a big role in quantum mechanics.  This program is based on describing as much of quantum mechanics as possible in terms of this kind of diagrammatic language.  This stuff has, in some ways, been explored more through the lens of computer science than physics per se – certainly Selinger is coming from that background.  There’s also more on this connection in the “Rosetta Stone” paper by John Baez and Mike Stay,
• My talk (actually third, but I put it here for logical flow) fits this framework, more or less.  I was in some sense there representing a viewpoint whose current form is due to Baez and Dolan, namely “groupoidification”.  The point is to treat the category $Span(Gpd)$ as a “categorification” of (finite dimensional) Hilbert spaces in the sense that there is a representation map $D : Span(Gpd) \rightarrow Hilb$ so that phenomena living in $Hilb$ can be explained as the image of phenomena in $Span(Gpd)$.  Having done that, there is also a representation of $Span(Gpd)$ into 2-Hilbert spaces, which shows up more detail (much more, at the object level, since Tannaka-Krein reconstruction means that the monoidal 2-Hilbert space of representations of a groupoid is, at least in nice cases, enough to completely reconstruct it).  This gives structures in $2Hilb$ which “conceptually” categorify the structures in $Hilb$, and are also directly connected to specific Hilbert spaces and maps, even though taking equivalence classes in $2Hilb$ definitely doesn’t produce these.  A “state” in a 2-Hilbert space is an irreducible representation, though – so there’s a conceptual difference between what “state” means in categorified and standard settings.  (There’s a bit more discussion in my notes for the talk than in the slides above.)
• Klaas Landsman was talking about what he calls “Bohrification“, which, on the technical side, makes use of Topos theory.  The philosophical point comes from Niels Bohr’s “doctrine of classical concepts” – that one should understand quantum systems using concepts from the classical world.  In practice, this means taking a (noncommutative) von Neumann algebra $A$ which describes the observables a quantum system and looking at it via its commutative subalgebras.  These are organized into a lattice – in fact, a site.  The idea is that the spectrum of $A$ lives in the topos associated to this site: it’s a presheaf that, over each commutative subalgebra $C \subset A$, just gives the spectrum of $C$.  This is philosophically nice in that the “Bohrified” propositions actually behave in a logically sensible way.  The topos approach comes from Chris Isham, developed further with Andreas Doring. (Note the series of four papers by both from 2007.  Their approach is in some sense dual to that of Lansman, Heunen and Spitters, in the sense that they look at the same site, but look at dual toposes – one of sheaves, the other of cosheaves.  The key bit of jargon in Isham and Doring’s approach is “daseinization”, which is a reference to Heidegger’s “Being and Time”.  For some reason this makes me imagine Bohr and Heidegger in a room, one standing on the ceiling, one on the floor, disputing which is which.)
• Gonzalo Reyes talked about synthetic differential geometry (SDG) as a setting for building general relativity.  SDG is a way of doing differential geometry in a category where infinitesimals are actually available, that is, there is a nontrivial set $D = \{ x \in \mathbb{R} | x^2 = 0 \}$.  This simplifies discussions of vector fields (tangent vectors will just be infinitesimal vectors in spacetime).  A vector field is really a first order DE (and an integral curve tangent to it is a solution), so it’s useful to have, in SDG, the fact that any differentiable curve is, literally, infinitesimally a line.  Then the point is that while the gravitational “field” is a second-order DE, so not a field in this sense, the arguments for GR can be reproduced nicely in SDG by talking about infinitesimally-close families of curves following geodesics.  Gonzalo’s slides are brief by necessity, but happily, more details of this are in his paper on the subject.

The other sessions I went to were mostly given by philosophers, rather than physicists or mathematicians, though with exceptions.  I’ll briefly present my own biased and personal highlights of what I attended.  They included sessions titled:

Quantum Physics“: Edward Slowik talked about the “prehistory of quantum gravity”, basically revisiting the debate between Newton and Leibniz on absolute versus relational space, suggesting that Leibniz’ view of space as a classification of the relation of his “monads” is more in line with relational theories such as spin foams etc.  M. Silberstein and W. Stuckey – gave a talk about their “relational blockworld” (described here) which talks about QFT as an approximation to a certain discrete theory, built on a graph, where the nodes of the graph are spacetime events, and using an action functional on the graph.

Meinard Kuhlmann gave an interesting talk about “trope bundles” and AQFTTrope ontology is an approach to “entities” that doesn’t assume there’s a split between “substrates” (which have no properties themselves), and “properties” which they carry around.  (A view of ontology that goes back at least to Aristotle’s “substance” and “accident” distinction, and maybe further for all I know).  Instead, this is a “one-category” ontology – the basic things in this ontology are “tropes”, which he defined as “individual property instances” (i.e. as opposed to abstract properties that happen to have instances).  “Things” then, are just collections of tropes.  To talk about the “identity” of a thing means to pick out certain of the tropes as the core ones that define that thing, and others as peripheral.  This struck me initially as a sort of misleading distinction we impose (say, “a sphere” has a core trope of its radial symmetry, and incidental tropes like its colour – but surely the way of picking the object out of the world is human-imposed), until he gave the example from AQFT.  To make a long story short, in this setup, the key entites are something like elementary particles, and the core tropes are those properties that define an irreducible representation of a $C^{\star}$-algebra (things like mass, spin, charge, etc.), whereas the non-core tropes are those that identify a state vector within such a representation: the attributes of the particle that change over time.

I’m not totally convinced by the “trope” part of this (surely there are lots of choices of the properties which determine a representation, but I don’t see the need to give those properties the burden of being the only ontologically primaries), but I also happen to like the conclusions because in the 2Hilbert picture, irreducible representations are states in a 2-Hilbert space, which are best thought of as morphisms, and the state vectors in their components are best thought of in terms of 2-morphisms.  An interpretation of that setup says that the 1-morphism states define which system one’s talking about, and the 2-morphism states describe what it’s doing.

New Directions Concerning Quantum Indistinguishability“: I only caught a couple of the talks in this session, notably missing Nick Huggett’s “Expanding the Horizons of Quantum Statistical Mechanics”.  There were talks by John Earman (“The Concept of Indistinguishable Particles in Quantum
Mechanics”), and by Adam Caulton (based on work with Jeremy Butterfield) on “On the Physical Content of the Indistinguishability Postulate”.  These are all about the idea of indistinguishable particles, and the statistics thereof.  Conventionally, in QM you only talk about bosons and fermions – one way to say what this means is that the permutation group $S_n$ naturally acts on a system of $n$ particles, and it acts either trivially (not altering the state vector at all), or by sign (each swap of two particles multiplies the state vector by a minus sign).  This amounts to saying that only one-dimensional representations of $S_n$ occur.  It is usually justified by the “spin-statistics theorem“, relating it to the fact that particles have either integer or half-integer spins (classifying representations of the rotation group).  But there are other representations of $S_n$, labelled by Young diagrams, though they are more than one-dimensional.  This gives rise to “paraparticle” statistics.  On the other hand, permuting particles in two dimensions is not homotopically trivial, so one ought to use the braid group $B_n$, rather than $S_n$, and this gives rise again to different statistics, called “anyonic” statistics.

One recurring idea is that, to deal with paraparticle statistics, one needs to change the formalism of QM a bit, and expand the idea of a “state vector” (or rather, ray) to a “generalized ray” which has more dimensions – corresponding to the dimension of the representation of $S_n$ one wants the particles to have.  Anyons can be dealt with a little more conventionally, since a 2D system may already have them.  Adam Caulton’s talk described how this can be seen as a topological phenomenon or a dynamical one – making an analogy with the Bohm-Aharonov effect, where the holonomy of an EM field around a solenoid can be described either dynamically with an interacting Lagrangian on flat space, or topologically with a free Lagrangian in space where the solenoid has been removed.

Quantum Mechanics“: A talk by Elias Okon and Craig Callender about QM and the Equivalence Principle, based on this.  There has been some discussion recently as to whether quantum mechanics is compatible with the principle that relates gravitational and inertial mass.  They point out that there are several versions of this principle, and that although QM is incompatible with some versions, these aren’t the versions that actually produce general relativity.  (For example, objects with large and small masses fall differently in quantum physics, because though the mean travel time is the same, the variance is different.  But this is not a problem for GR, which only demands that all matter responds dynamically to the same metric.)  Also, talks by Peter Lewis on problems with the so-called “transactional interpretation” of QM, and Bryan Roberts on time-reversal.

Why I Care About What I Don’t Yet Know“:  A funny name for a session about time-asymmetry, which is the essentially philosophical problem of why, if the laws of physics are time-symmetric (which they approximately are for most purposes), what we actually experience isn’t.  Personally, the best philosophical account of this I’ve read is Huw Price’s “Time’s Arrow“, though Reichenbach’s “The Direction of Time” has good stuff in it also, and there’s also Zeh’s more technical “The Physical Basis of the Direction of Time“. In the session, Chris Suhler and Craig Callender gave an account of how, given causal asymmetry, our subjective asymmetry of values for the future and the past can arise (the intuitively obvious point being that if we can influence the future and not the past, we tend to value it more).  Mathias Frisch talked about radiation asymmetry (the fact that it’s equally possible in EM to have waves converging on a source than spreading out from it, yet we don’t see this).  Owen Maroney argued that “There’s No Route from Thermodynamics to the Information Asymmetry” by describing in principle how to construct a time-reversed (probabilisitic) computer.  David Wallace spoke on “The Logic of the Past Hypothesis”, the idea inspired by Boltzmann that we see time-asymmetry because there is a point in what we call the “past” where entropy was very low, and so we perceive the direction away from that state as “forward” it time because the world tends to move toward equilibrium (though he pointed out that for dynamical reasons, the world can easily stay far away from equilibrium for a long time).  He went on to discuss the logic of this argument, and the idea of a “simple” (i.e. easy-to-describe) distribution, and the conjecture that the evolution of these will generally be describable in terms of an evolution that uses “coarse graining” (i.e. that repeatedly throws away microscopic information).

The Emergence of Spacetime in Quantum Theories of Gravity“:  This session addressed the idea that spacetime (or in some cases, just space) might not be fundamental, but could emerge from a more basic theory.  Christian Wüthrich spoke about “A-Priori versus A-Posteriori” versions of this idea, mostly focusing on ideas such as LQG and causal sets, which start with discrete structures, and get manifolds as approximations to them.  Nick Huggett gave an overview of noncommutative geometry for the philosophically minded audience, explaining how an algebra of observables can be treated like space by means of all the concepts from geometry which can be imported into the theory of $C^{\star}$-algebras, where space would be an approximate description of the algebra by letting the noncommutativity drop out of sight in some limit (which would be described as a “large scale” limit).  Sean Carroll discussed the possibility that “Space is Not Fundamental – But Time Might Be”, pointing out that even in classical mechanics, space is not a fundamental notion (since it’s possible to reformulate even Hamiltonian classical mechanics without making essential distinctions between position and momentum coordinates), and suggesting that space arises from the dynamics of an actual physical system – a Hamiltonian, in this example – by the principle “Position Is The Thing In Which Interactions Are Local”.  Finally, Sean Maudlin gave an argument for the fundamentality of time by showing how to reconstruct topology in space from a “linear structure” on points saying what a (directed!) path among the points is.

It’s been a while since I posted last, but in there I described some issues related to talks I gave in Portugal recently. I’m beginning a postdoc at the Instituto Superior Tecnico, in Lisbon, in less than a month’s time. In the meantime, I’ve been two weeks in Portugal, including a conference and apartment hunting.  Then, last week, I got married. So not surprisingly, I’ve been a bit slow in updating.

The talks I gave are this one, which I gave at IST and this one at the XIX Oporto Meeting on Geometry, Topology and Physics which was held this year in Faro, which this year was a conference on the theme of Categorification!  These talks also appear on my new website, which I got because my hosting at UWO will expire sooner or later, and I wanted something portable (and a portable email address came with it).

## Lisbon

Lisbon is an interesting city.  I’ve visited Europe before for conferences and travel and so on, but never for long, and have only lived in North America, where most urban areas are much newer and ancient history more poorly documented.  This is even more so in the southern parts of Europe that were part of the Roman Empire (and even more so in areas of India I’ve travelled in).  I’m looking forward to getting more familiar with the place, which has an exciting and under-appreciated history.  At least I assume it’s underappreciated, since a majority of people in Canada who ask me where I’m moving have never even heard of Lisbon, which I find surprising.

Human settlement in Portugal actually pre-dates homo sapiens, going back to Neanderthals (we often forget there’ve been a few dozen human species before ours. and our era is unusual in human history for having just the one).  Among Sapiens, there have been various periods, most recently the ancient megalith-building cultures, Phoenecians, Greeks, Carthaginians, Romans, Visigoths, Arabs, and then the kingdom (now a republic) of Portugal, established during the Christian reconquest of Iberia.  Lisbon itself dates back at least to Roman times. The oldest surviving areas of Lisbon date back (in streetplan, if not actual buildings) to the Moorish kingdom, when Iberia was known as al-Andalus, some 800 years ago.  Lisbon’s downtown, immediately below this area, couldn’t be more different, being one of the first urban areas planned on a grid – this followed the original area being destroyed in an earthquake and resulting tsunami in 1755.  As the capital of Europe’s first overseas empire, which had reached Japan and Brazil by well over 400 years ago, Lisbon has been a “global city” for at least that long, with spells of boom and bust, and more recently, dictatorship and revolution.  Its location means it was historically a hub that linked the older Mediterranean trading world and the larger Atlantic and Indian Ocean world.

Here is a picture of the main pavillion on the IST campus:

And here is a picture of the neighborhood where I’ll be living, about 10-15 minutes’ walk or two metro stops away:

As you can also see from these pictures, Lisbon contains a number of hills.  It is occasionally reminiscent of San Francisco in that way, and the style of buildings, which also resembles New Orleans occasionally.  And of course, since this is Europe, public spaces that look like this:

And so on.

## Visit at IST

Anyway, in the visit at IST, we also had a little mini-conference on categorification, featuring some people who also spoke at Faro (including me) giving longer and more elaborate versions of our talks.  I already commented on mine, so I’ll mention the others:

Rafael Diaz gave a talk about how to categorify noncommutative or “quantum” algebras, in the sense of algebras of power series in noncommuting variables, using ideas similar to the way commutative polynomial algebras can be “categorified” by Joyal’s species.  This “quantum species” idea is laid out partially in this paper. This leads on into ideas about categorifying deformation quantization.

The basic point is to think of “a categorification of a ring R” as a distributive category $(C,\oplus,\otimes)$ whose Burnside ring (the ring of isomorphism classes of objects, with algebraic operations from $\oplus$ and $\otimes$) is $R$, or more generally has a “valuation” valued in $R$ that is surjective and gets along with the algebra operations.

The category chosen to describe a deformation of $R$ is then the category of functors from $FinSet_0^k$ into $C$.  The main point is then to find a noncommutative product operation $\star$, in place of the obvious one derived from $\otimes$, which gives a categorification of a polynomial ring.  This has to do with sticking structured sets together, where some elements of the set can form “links” between the elements of sets – this uses three-“coloured” sets, where one “colour” denotes elements associated to links.

Yazuyoshi Yonezawa gave a talk about some stuff related to link homology invariants such as Khovanov homology.  Such invariants are a major theme for people interested in categorification these days, for various specific reasons, but in general because tangle categories have some nice universal properties, so doing certain kinds of universal higher-dimensional algebra naturally has applications to studying tangles, hence links, hence knots.  In particular, invariants like the Jones, HOMFLY, and HOMFLYPT polynomials, and Reshitikhin-Turaev invariants.  Yazuyoshi’s talk was about an approach to these things based on – as I understood it – some representation theory of quantum $\mathfrak{sl}_n$, and a diagrammatic calculus that goes with it, for assigning data to strands and crossings of a knot.  (This sort of thing gives a knot invariant as long as it’s invariant under Reidemeister moves – that is, is unaffected by changing the presentation of the knot.  Many of the knot invariants that come up here arise from treating the knot using some sort of diagrammatic calculus – which is where the category theory comes in.)

Aleksandar Mikovic gave a talk about higher gauge theory in the form of 2-BF theory – also known as BFCG theory, this is sort of the “categorified” equivalent of the theory of a flat connection, now taking values in a Lie 2-group.  Actually, he speaks about these in terms of Lie crossed modules, which is a rather nice language for talking about higher-algebraic group-like gadgets in terms of chains of groups with some extra structure (actions of lower groups on higher, and some other things) – see Tim Porter’s “Crossed Menagerie” for a comprehensive look.  The talk was related to finding gauge invariant actions for theories of this sort – the paper it’s based on is one with Joao Faria Martins.

## XIX Oporto Meeting

The Oporto meeting on geometry and physics, specifically devoted this year to categorification, was very interesting, with a range of good speakers. Unfortunately, Faro is not optimal as a conference site: the accomodations are a half-hour bus ride from the campus where the conference is held, and the buses come only about once per hour and as a result (of that, and jet-lag, which could happen anyway), I missed some of the talks. Otherwise, it’s a pleasant town with a nice atmosphere, and it was interesting to see some of the variety of people working on categorification.  In particular, a lot of people are working on categorifying aspects of representation theory, which in turn is interesting to topologists, and knot theorists in particular.

One bunch of ideas about categorical representations which was referred to a lot is due to Chuang and Rouquier, substantially described in a paper from a few years ago – here is a post from the Secret Blogging Seminar a few years back describing some of the ideas a bit more succinctly.  The basis for the most popular program being discussed, and the big idea in recent years, is due to Khovanov and Lauda – see the bottom section of this post.

Now, the main invited speakers each gave a series of three hour-long classes on their topic in the mornings, while in the afternoons the other speakers gave 20-minute talks.  The main speakers were these:

Mikhail Khovanov wasn’t able to attend for personal reasons, but there was a great deal of discussion about work that builds on his categorification of quantum groups with Aaron Lauda, who however was there and gave a nice series of talks introducing the ideas (though I missed some because of the unfortunate bus infrastructure). Aaron collects a bunch of resources on this subject here, and I’ll explain a bit of this below.

Sabin Cautis talked about the categorification of $sl_2$ in terms of geometric representation theory; the idea here is that there are certain spaces that carry natural representations.  These are flag varieties – the simplest example being Grassmanians – spaces whose points are the $k$-dimensional subspaces of some fixed $V$. In general, flag varieties are spaces whose points consist of a nested sequence of subspaces $V_0 \subset V_1 \subset \dots \subset V_k = V$ (the terminology “flag” suggests a flagpole with a 2D rectangle, suspended from a 1D pole, rooted at a 0D point).  The talk was an overview of how to use this to categorify some representation theory.  Here is a recent related paper by Cautis, including Joel Kamnitzer, (I blogged his talk here at UWO a while ago on a similar subject in some more detail), and Anthony Licata.  The basic point is that categories of sheaves on such spaces carry a categorical representation of $\mathfrak{sl}_2$.

One thing I found interesting – this time, as with Joel’s talk, is that span constructions turn up in this stuff quite naturally, but there is both a similarity and a difference in how they’re used.  In particular, given a flag $V_0 \subset \dots \subset V_i \subset \dots \subset V_k$, we can project to a flag with one fewer entries just by omitting $V_i$.  So the various flag varieties associated to $V$ are connected by a bunch of projections.  Taking two different projections (dropping, say $V_i$ and $V_j$), we get a span of varieties – that is, one object with two maps out of it.  We’re talking about spaces of functions on these varieties, so pushing these through spans is of interest.  Lifting a function (by pre-composition – assign a flag the value of the function at its image) is easy – pushing forward is harder.  This involves taking a sum over the function values over the preimage – all the long flags that project to a given short one (to make sure this is tractable, we consider only constructible functions, with finitely many values).  But this sum is weighted.  In the groupoidification program, something similar happens, but the weight there is the groupoid cardinality of the preimage.  Here, it is the Euler characteristic of the preimage (or rather, for each function value, the part of the preimage taking a given value contributes its Eular char. as the weight for that value).  Since groupoid cardinality is like a multiplicative sort of Euler characteristic, there seems to be a close analog here I’d like to understand better.

Catharina Stroppel talked about how the subject relates to Soergel bimodules, and led up to categorifying 3j-symbols.  Soergel bimodules showed up in several different talks about this stuff.  These are the irreducible summands in the bimodule that comes from applying induction functions between module categories $Ind: R^{\lambda'}-mod \rightarrow R^{\lambda}-mod$ finitely many times.

Here, the $R^{\lambda}$ are  rings of functions invariant under $S_{\lambda}$, which is the subgroup of the permutation group $S_n$ which respects a particular composition $\lambda$ of $n$ (like a partition, but with order – compositions also specify flag varieties, by specifying the codimensions at each inclusion).  The point is that, if $S_{\lambda'} < S_{\lambda}$, we get inclusions of the rings of invariant functions, and then we can induce representations along those inclusions.  (Notice, by the way, that the correspondence between compositions and the signature of a flag means that this is actually much the same as the inclusions I just described under Sabin Cautis’ talk).  Then doing a finite chain of such inductions gives a functor between module categories.  This can be described by tensoring with some $(R^{\lambda'},R^{\lambda})$ bimodule – the direct summands in this are the Soergel bimodules.  So these are central in talking about these categorical actions and categorified representation theory.  This in turn ended up, in this series of talks, at a categorification of 3j-symbols (which can be built using representations and intertwiners).

Ben Webster talked about how diagrammatic methods used in the Khovanov-Lauda program can be used to categorify algebra representations, and through that, the Reshitikhin-Turaev invariant; the key diagrammatic element turns out to be marking special “red” lines with special rules allowing strands to “act” on them by concatenation.  I must admit Ben Webster’s talks, which ended up rather technical, went far enough over my head that I’m reluctant to summarize, since I was still catching up on the KL program, and this was carrying it quite a bit further.  I do recall that there was much discussion of cyclotomic quotients (partly because Alex Hoffnung later came back to the matter and I had a chance to talk to him about it briefly) – that is, the quotients imposing the relations forcing something to be a root of unity, which isn’t surprising since quantum groups at $q$ a root of unity are important and special.  Luckily for the reader who is more up on this stuff than I, the slides can be found here and here.

Dylan Thurston spoke on Heegard-Floer homology (slides here, here, and here – full of great pictures, by the way), which is a homology theory for 3-manifolds (then an invariant for a closed 4-manifold), due to Oszvath and Szabo.  It’s a bi-graded homology theory (i.e. homology theories give complexes for spaces – this gives a bicomplex, with grading in two directions).  This theory gives back the (Conway-)Alexander polynomial for a knot when you take the Euler characteristic of the bicomplex.  That is: there are two directions this complex is graded in: one (columns, say) will correspond to the degree of the variable $t$ in the Alexander polynomial; for each $k$, the coefficient of $t^k$ is the Euler characteristic (alternating sum of dimensions) of the entries in that column.  So this is a categorification of this polynomial, in somewhat the way that Khovanov homology categorifies the Jones polynomial.

HF homology can be defined for a knot can be defined in a combinatorial way: a 3-manifold can be represented by a “Heegard diagram” – a 2D surface marked with (coloured) curves, which is a way of keeping track of how a 3-manifold is built by splitting it into parts.  From this diagram, one gets “grid diagrams”, and by a combinatorial process (see the slides for more details) generates the complex.

Others.  I didn’t manage to attend all the other talks (partly because of aforementioned bus issues, and partly because I was still working on mine, having taken a lot of time in Lisbon doing useless things like finding a place to live), but among those I did, there were several that were based on the Khovanov-Lauda program for categorified quantum groups: Anna Beliakova in particular worked with them on categorifying the Casimir (generator of the centre) of the categorified quantum group; people working with Soergel bimodules and categorified Hecke algebras such as Ben Elias and Nicholas Liebedinski.  Then there were the connections to link homology: Christian Blandet and Geordie Williamson talked about things related to the HOMFLYPT polynomial; Krystof Putyra and Emmanuel Wagner gave talks related to Khovanov homology and link homology.  Alex Hoffnung talked about a combinatorial approach for dealing with categorification of cyclotomic quotients as discussed by Ben Webster.

## Categorification of Quantum Groups

The reason for categorifying quantum groups, at least in this context, has to do with the manifold invariants associated to them.  Often these come from categories of representations of groups or quantum groups – more generally ones with similar formal properties, meaning roughly monoidal categories with duals (and possibly more structure).  These give state sum invariants, by assigning data from the category to a triangulation of a manifold – objects on edges and morphisms on triangles, say.   The categorification of quantum groups means we pass from having a monoidal category with duals (of representations), to a monoidal 2-category with duals (of representations).  This would mean the state-sum invariants it’s natural to construct are now for 4-manifolds, rather than 3-manifolds.  This is the premise behind spin foam models in gravity, but also has its own life within quantum topology as tools for classifying manifolds, whether or not it accurately describes anything physical.  Marco Mackaay, one of the conference organizers (among several others), has written a bunch on this – for example, this constructs a state-sum invariant given any “spherical” 2-category (a property of certain monoidal 2-categories – see inside for details), and this gives a specific consstruction using the Khovanov-Lauda categorification of $\mathfrak{sl}_3$.

The Khovanov-Lauda approach to categorifying quantum groups (in particular, deformations of envelopoing algebras of classical Lie algebras, within the category of Hopf algebras)  is most basically about “categorifying” the presentation of an algebra in terms of generators and relations.  That is, we describe a set with some operations in terms of some elements of the set (generators), and some equations (relations) which they satisfy involving the operations.  The presentation used for $U_q(\mathfrak{sl}_n)$ is the standard one based on an n-vertex (type-A) Dynkin diagram: basically, $n$ dots in a row.  There’s a generator $e_i$ for the $i^{th}$ vertex; the generators for non-adjacent vertices all commute, and for adjacent generators, we have $(q + q^{-1}) e_i e_j e_i = e_j e_i e_j$.  (The factor involving $q$ is the quantum integer $[2]_q$, and becomes 2 in the limit).

To categorify this, we still give generators, but the equations are replaced with isomorphisms – this means we need to be working in some category $R$, hence one essential task is to describe the morphisms.  So: the objects are just rows of dots, labelled by vertices of the Dynkin diagram.  The morphisms are (linearly generated by) isotopy classes of braids from one row to another.  The essential thing is that we have to carefully define “isotopy” here to ensure we get the categorified version of the relations above.  So for non-adjacent-vertex labels, we have the usual Reidemeister moves (the key ones being: we can slide a strand past a crossing, straighten out two complementary crossings); for adjacent-vertex labels, though, we have to tweak this, imposing some relations on strands involving the factors of $q$.  The relations take up a few slides in the talk, but essentially are chosen so that:

Theorem (Khovanov-Lauda): There is an isomorphism of twisted bialgebras between the positive part of $U_q(\mathfrak{sl}_n)$ and the Grothendieck ring $K_0(R)$, where multiplication and comultiplication are given by the image of induction and restriction.

Obviously, much more could be said from a five-day conference, but this seems like a nice punchline.

It’s taken me a while to write this up, since I’ve been in the process of moving house – packing and unpacking and all the rest. However, a bit over a week ago, I was in Montreal, attending MakkaiFest ’09 at the Centre de Recherches Mathematiques at the University of Montréal (and a pre-conference workshop hosted at McGill, which I’m including in the talks I mention here). This was in honour of the 70th birthday of Mihaly (Michael) Makkai, of McGill University. Makkai has done a lot of important foundational work in logic, model theory, and category theory, and a great many of the talks were from former students who’d gone on and been inspired by him, so one got sense of the range of things he’s worked on through his life.

The broad picture of Makkai’s work was explained to us by J.P. Marquis, from the Philosophy department at U of M. He is interested in philosophy of mathematics, and described Makkai’s project by contrast with the program of axiomatization of the early 20th century, along the lines suggested by Hilbert. This program provided a formal language for concrete structures – the problem, which category theory is part of a solution to, is to do the same for abstract structures. Contrast, for instance, the concrete description of a group $G$ as a (particular) set with some (particular) operation, with the abstract definition of a group object in a category. Makkai’s work in categorical logic, said Marquis, is about formalizing the process of abstraction that example illustrates.

Model Theory/Logic

This matter – of the relation between abstract theories and concrete models of the theories – is really what model theory is about, and this is one of the major areas Makkai has worked on. Roughly, a theory is most basically a schema with symbols for types, members of types, and some function symbols – and a collection of sentences built using these symbols (usually generated from some axioms by rules of logical inference). A model is (intuitively), an interpretation of the terms: a way of assigning concrete data to the symbols – say, a symbol for a type is assigned the set of all entities of that type, and a function symbol is assigned an actual function between sets, and so on – making all propositions true. A morphism of models is a map that preserves all the properties of the model that can be stated using first order logic.

This is an older way to say things – Victor Harnik gave an expository talk called “Model Theory vs. Categorical Logic” in which he compared two ways of adding an equivalence relation to a theory. The model theory way (invented by Shelah) involves taking the theory (list of sentences) $T$ and extending it to a new theory $T^{eq}$. This has, for instance, some new types – if we had a type for “element of group”, for example, we might then get a new type “equivalence class of elements of group”, and so on. Now, this extension is “tight” in the sense that the categories of all models of $T$ and of $T^{eq}$ are equivalent (by a forgetful functor $Mod(T^{eq}) \rightarrow Mod(T)$) – but one can prove new theorems in the extended theory. To make this clear, he described work (due to Makkai and Reyes) about pretopos completion. Here, one has the concept of a “Boolean logical category” – $Set$ is an example, as is, for any theory, a certain category whose objects are the formulas of the theory. This is related to Lawvere theories (see below). There are logical functors between such categories – functors into $Set$ are models, but there are also logical functors between theories. The point is that a theory $T$ embeds into $T^{eq}$ (abusing notation here – these are now the boolean logical categories). Then the point is that $T^{eq}$ arises as a kind of completion of $T$ – namely, it’s a boolean pretopos (not just category). Moreover, it has some nice universal properties, making this point of view a bit more natural than the model-theoretic construction.

Bradd Hart’s talk, “Conceptual Completeness for Cantinuous Logic”, was a bit over my head, but made some use of this kind of extension of a theory to $T^{eq}$. The basic point seems to be to add some kind of continuous structure to logic. One example comes from a metric structure – defining a metric space of terms, where the metric function $d(x,y)$ is some sum $\sum_n \phi_n (x,y)$, where the $\phi_n$ are formulas with two variables, either true or false – where true gives a $0$, and false gives a $1$ in this sum. This defines a distance from $x$ to $y$ associated to the given list of formulas $\phi_n$. A continuous logic is one with a structure like this. The business about equivalence relations arises if we say two things are equivalent when the distance between them is 0 – this leads to a concept of completion, and again there’s a notion that the categories of models are equivalent (though proving it here involves some notion of approximating terms to arbitrary epsilon, which doesn’t appear in standard logic).

Anand Pillay gave a talk which used model theory to describe some properties of the free group on n generators. This involved a “theory of the free group” which applies to any free group, and regard each such group as a model of the theory – in fact a submodel of some large model, and using model-theoretic methods to examine “stability” properties, in some sense which amounts to a notion of defining “generic” subsets of the group.

Logic and Higher Categories

A number of talks specifically addressed the ground where logic meets higher dimensional categories, since Makkai has worked with both.

In one talk, Robert Paré described a way of thinking about first-order theories as examples of “double Lawvere theories”. Lawvere’s way of formalizing “theories and models” was to say that the theory is a category itself (which has just the objects needed to describe the kind of structure it’s a theory of) – and a model is a functor into $Sets$ (or some other category – a model of the theory of groups in topological spaces, say, is a topological group). For example, the theory of groups includes an object $G$ and powers of it, multiplication and inverse maps, and expresses the axioms by the fact that certain diagrams commute. A model is a functor $M : Th(Grp) \rightarrow Sets$, assigning to the “group object” a set of elements, which then get the group structure from the maps. Instead of a category, this uses a double category. There are two kinds of morphisms – horizontal and vertical – and these are used to represent two kinds of symbols: function symbols, and relation symbols. (For example, one can talk about the theory of an ordered field – so one needs symbols for multiplication and addition and so forth, but also for the order relation $\leq$). Then a model of such a theory is a double functor into the double category whose objects are sets, and whose horizontal and vertical morphisms are respectively functions and relations.

André Joyal gave a talk about the first order logic of higher structures. He started by commenting on some fields which began life close together, and are now gradually re-merging: logic and category theory; category theory and homotopy theory (via higher categories); homotopy theory and algebraic geometry. The higher categories Joyal was thinking of are quasicategories, or “$( \infty, 1)$-categories, which are simplicial sets satisfying a weak version of a horn-filling condition (the “strict” version of this, a Kan complex, includes as example $N(C)$, the nerve of a category $C$ – there’s an n-simplex for each sequence of n composable morphisms, whose other edges are the various composites, and whose faces are “compositors”, “associators”, and so on – which for $N(C)$ are identities). The point of this is that one can reproduce most of category theory for quasicategories – in particular, he mentioned limits and colimits, factorization systems, pretoposes, and model theory.

Moving to quasicategories on one side of the parallel between category theory and logic has a corresponding move on the other side – on the logic side, one aspect is that the usual notion of a language is replaced by what’s called Martin-Löf type theory. This, in fact, was the subject of Michael Warren’s talk, “Martin-Löf complexes” (I reported on a similar talk he gave at Octoberfest last year). The idea here is to start by defining a globular set, given a theory and type $A$ – a complex whose n-cells have two faces, of dimension (n-1). The 0-cells are just terms of some type $A$. The 1-cells are terms of types like $\underline{A}(a,b)$, where $a$ and $b$ are variables of type $A$ – the type has an interpretation as a proposition that $a=b$ “extensionally” (i.e. not via a proof – but as for instance when two programs with non-equivalent code happen to always produce the same output). This kind of operation can be repeated to give higher cells, like $\underline{A(a,b)}(f,g)$, and so on. Given a globular set $G$, one gets a theory by an adjoint construction. Putting the two together, one has a monad on the category of globular sets – algebras for the monad are Martin-Löf complexes. Throwing in syntactic rules to truncate higher cells (I suppose by declaring all cells to be identities) gives n-truncated versions of these complexes, $MLC_n$. Then there is some interesting homotopy theory, in that the category of n-truncated Martin-Löf complexes is expected to be a model for homotopy n-types. For example, $MLC_0$ is equivalent to $Sets$, and there is an adjunction (in fact, a Quillen equivalence – that is, a kind of “homotopy” equivalence) between $MLC_1$ and $Gpd$.

Category Theory/Higher Categories

There were a number of talks that just dealt with categories – including higher categories – in their own right. Makkai has worked, for example, on computads, which were touched on by Marek Zawadowski in one of his two talks (one in the pre-conference workshop, the other in the conference). The first was about categories of “many-to-one shapes”, which are important to computads – these are a notion of higher-category, where every cell takes many “input” faces to one “output” face. Zawadowski described a “shape” of an n-cell as an initial object in a certain category built from the category of computads with specified faces. Then there’s a category of shapes, and an abstract description of “shape” in terms of a graded tensor theory (graded for dimension, and tensor because there’s a notion of composition, I believe). Zawadowski’s second talk, “Opetopic Sets in Lax Monoidal Fibrations”, dealt with a similar topic from a different point of view. A lax monoidal fibration (LMF) is a kind of gadget for dealing with multi-level structures (categories, multicategories, quasicategories, etc). There’s a lot of stuff here I didn’t entirely follow, but just to illustrate: categories arise as LMF, by the fibration $cod : Set^{B} \rightarrow Set$, where $B$ is the category with two objects $M, O$, and two arrows from $M$ to $O$. An object in the functor category $Set^{B}$ consists of a “set of morphisms and set of objects” with maps – making this a category involves the monoidal structure, and how composition is defined, and the real point is that this is quite general machinery.

Joachim Lambek and Gonzalo Reyez, both longtime collaborators and friends of Makkai, also both gave talks that touched on physics and categories, though in very different ways. Lambek talked about the “Lorentz category” and its appearance in special relativity.  This involves a reformulation of SR in terms of biquaternions: like complex numbers, these are of the form $u + iv$, but $u$ and $v$ are quaternions.  They have various conjugation operations, and the geometry of SR can be described in terms of their algebra (just as, say, rotations in 3D can be described in terms of quaternions).  The Lorentz category is a way of organizing this – its two objects correspond to “unconjugated” and “conjugated” states.

Gonzalo Reyez gave a derivation of General Relativity in the context of synthetic differential geometry.  The substance of this derivation is not so different from the usual one, but with one exception.  Einstein’s field equations can be derived in terms of the motions of small regions full of of freely falling test particles – synthetic differential geometry makes it possible to do the same analysis using infinitesimals rigorously all the way through.  The basic point here is that in SDG one replaces the real line as usually conceived, with a “real line with infinitesimals” (think of the ring $\mathbb{R}[\epsilon]/\langle \epsilon^2 \rangle$, which is like the reals, but has the infinitesimal $\epsilon$, whose square is zero).

Among other talks: John Power talked about the correspondence between Lawvere theories in universal algebra and finitary tree monads on sets – and asked about what happens to the left hand side of this correspondence when we replace “sets” with other categories on the righ hand side. Jeff Egger talked about measure theory from a categorical point of view – namely, the correspondence of NCG between C*-algebras and “noncommutative” topological spaces, and between W*-algebras and “noncommutative” measure spaces, thought of in terms of locales. Hongde Hu talked about the “codensity theorem”, and a way to classify certain kinds of categories – he commented on how it was inspired by Makkai’s approach to mathematics: 1) Find new proofs of old theorems, (2) standardize the concepts used in them, and (3) prove new theorems with those concepts. Fred Linton gave a talk describing Heath’s “V-space”, which is a half-plane with a funny topology whose open sets are “V” shapes, and described how the topos of locally finite sheaves over it has surprising properties having to do with nonexistence of global sections. Manoush Sadrzadeh, whom I met recently at CQC (see the bottom of the previous post) was again talking about linguistics using monoidal categories – she described some rules for “clitic movement” and changes in word order, and what these rules look like in categorical terms.

Other

A few other talks are a little harder for me to fit into the broad classification above.  There was Charles Steinhorn’s talk about ordered “o-minimal” structures, which touched on a bit of economics – essentially, a lot of economics is based on the assumption that preference orders can be made into real-valued functions, but in fact in many cases one has (variants on) “lexicographic order”, involving ranked priorities.  He talked about how typically one has a space of possibilities which can be cut up into cells, with one sort of order in each cell.  There was Julia Knight, talking about computable structures of “high Scott rank” – in particular, this is about infinite structures that can still be dealt with computably – for example, infinitary logical formulas involving an infinite number of “OR” statements where all the terms being joined are of some common form.  This ends up with an analysis of certain infinite trees.  Hal Kierstead gave a talk about Ramsey theory which I found notable because it used the kind of construction based on a game: to prove that any colouring of a graph (or hypergraph) has some property, one devises a game where one player tries to build a graph, and the other tries to colour it, and proves a winning strategy for one player.  Finally, Michael Barr gave a talk about a duality between certain categories of modules over commutative rings.

All in all, an interesting conference, with plenty of food for thought.

Barr, Kierstead, Knight, Steinhorn

Continuing from the previous post…

I realized I accidentally omitted Klaas Lansdman’s  talk on the Kochen-Specker theorem, in light of topos theory.  This overlaps a lot with the talk by Andreas Doring, although there are some significant differences.  (Having heard only what Andreas had to say about the differences, I won’t attempt to summarize them).  Again, the point of the Kochen-Specker theorem is that there isn’t a “state space” model for a quantum system – in this talk, we heard the version saying that there are no “locally sigma-Boolean” maps, from operators on a Hilbert space, to $\{ 0, 1 \}$.  (This is referring to sigma-algebas (of measurable sets on a space), and Boolean algebras of subsets – if there were such a map, it would be representing the system in terms of a lattice equivalent to some space).  As with the Isham/Doring approach, they then try to construct something like a state space – internal to some topos.  The main difference is that the toposes are both categories of functors into sets from some locale – but here the functors are covariant, rather than contravariant.

Now, roughly speaking, the remaining talks could be grouped into two kinds:

Quantum Foundations

Many people came to this conference from a physics-oriented point of view.  So for instance Rafael Sorkin gave a talk asking “what is a quantum reality?”. He was speaking from a “histories” interpretation of quantum systems. So, by contrast, a “classical reality” would mean one worldline: out of some space of histories, one of them happens. In quantum theory, you typically use the same space of histories, but have some kind of “path integral” or “sum over histories” when you go to compute the probabilities of given events happening. In this context, “event” means “a subset of all histories” (e.g. the subset specified by a statement like “it rained today”). So his answer to the question is: a reality should be a way of answering all questions about all events.  This is called a “coevent”.  Sorkin’s answer to “what is a quantum reality?” is: “a primitive, preclusive coevent”.

In particular, it’s a measure $\mu$.  For a classical system, “answering” questions means yes/no, whether the one history is in a named event – for a quantum system, it means specifying a path integral over all events – i.e. a measure on the space of events.  This measure needs some nice properties, but it’s not, for instance, a probability measure (it’s complex valued, so there can be interference effects).  Preclusion has to do with the fact that the measure of an event being zero means that it doesn’t happen – so one can make logical inferences about which events can happen.

Other talks addressing foundational problems in physics included Lucien Hardy’s: he talked about how to base predictive theories on operational structures – and put to the audience the question of whether the structures he was talking about can be represented categorically or not.  The basic idea is an “operational structure” is some collection of operations that represents a physical experiment whose outcome we might want to predict.  They have some parameters (“knob settings”), outcomes (classical “readouts”), and inputs and outputs for the things they study and affect (e.g. a machine takes in and spit out an electron, doing something in the middle).  This sort of thing can be set up as a monoidal category – but the next idea, “object-oriented operationalism”, involved components having “connections” (given relations between their inputs) and “coincidences” (predictable correlations in output).  The result was a different kind of diagram language for describing experiments, which can be put together using a “causaloid product” (he referred us to this paper, or a similar one, on this).

Robert Spekkens gave a talk about quantum theory as a probability theory – there are many parallels, though the complex amplitudes give QM phenomena like interference.  Instead of a “random variable” $A$, one has a Hilbert space $H_A$; instead of a (positive) function of $A$, one has a positive operator on $H_A$; standard things in probability have analogs in the quantum world.  What Robert Spekkens’ talk dealt with was how to think about conditional probabilities and Bayesian inference in QM.  One of the basic points is that when calculating conditional probabilities, you generally have to divide by some probability, which encounters difficulties translating into QM.  He described how to construct a “conditional density operator” along similar lines – replacing “division” by a “distortion” operation with an analogous meaning.  The whole thing deeply uses the Choi-Jamiolkowski isomorphism, a duality between “states and channels”.  In terms of the string diagrams Bob Coecke et. al. are keen on, this isomorphism can be seen as taking a special cup which creates entangled states into an ordinary cup, with an operator on one side.  (I.e. it allows the operation to be “slid off” the cup).  The talk carried this through, and ended up defining a quantum version of the probabilistic concept of “conditional independence” (i.e. events A and C are independent, given that B occurred).

A more categorical look at foundational questions was given by Rick Blute’s talk on “Categorical Structures in AQFT”, i.e. Algebraic Quantum Field Theory.  This is a formalism for QFT which takes into account the causal structure it lives on – for example, on Minkowski space, one has a causal order for points, with $x \leq y$ if there is a future-directed null or timelike curve from $x$ to $y$.  Then there’s an “interval” (more literally, a double cone) $[x,y] = \{ z | x \leq z \leq y\}$, and these cones form a poset under inclusion (so this is a version of the poset of subspaces of a space which keeps track of the causal structure).  Then an AQFT is a functor $\mathbb{A}$ from this poset into C*-algebras (taking inclusions to inclusions): the idea is that each local region of space has its own algebra of observables relevant to what’s found there.  Of course, these algebras can all be pieced together (i.e. one can take a colimit of the diagram of inclusions coming from all regions on spacetime.  The result is $\hat{\mathbb{A}}$.  Then, one finds a category of certain representations of it on a hilbert space $H$ (namely, “DHR” representations).  It turns out that this category is always equivalent to the representations of some group $G$, the gauge group of the AQFT.  Rick talked about these results, and suggested various ways to improve it – for example, by improving how one represents spacetime.

The last talk I’d attempt to shoehorn into this category was by Daniel Lehmann.  He was making an analysis of the operation “tensor product”, that is, the monoidal operation in $Hilb$.  For such a fundamental operation – physically, it represents taking two systems and looking at the combined system containing both – it doesn’t have a very clear abstract definition.  Lehmann presented a way of characterizing it by a universal property analogous to the universal definitions for products and coproducts.  This definition makes sense whenever there is an idea of a “bimorphism” – a thing which abstracts the properties of a “bilinear map” for vector spaces.  This seems to be closely related to the link between multicategories and monoidal categories (discussed in, for example, Tom Leinster’s book).

Categories and Logic

Some less physics-oriented and more categorical talks rounded out the part of the program that I saw.  One I might note was Mike Stay‘s talk about the Rosetta Stone paper he wrote with John Baez.  The Rosetta Stone, of course, was a major archaeological find from the Ptolemaic period in Egypt – by that point, Egypt had been conquered by Alexander of Macedon and had a Greek speaking elite, but the language wasn’t widespread.  So the stone is an official pronouncement with a message in Greek, and in two written forms of Egyptian (heiroglyphic and demotic), neither of which had been readable to moderns until the stone was uncovered and correspondences could be deduced between the same message in a known language and two unknown ones.  The idea of their paper, and Mike’s talk, is to collect together analogs between four subjects: physics, topology, computation, and logic.  The idea is that each can be represented in terms of monoidal categories.  In physics, there is the category of Hilbert spaces; in topology one can look at the category of manifolds and cobordisms; in computation, there’s a monoidal category whose objects are data types, and whose morphisms are (equivalence classes) of programs taking data of one type in and returning data of another type; in logic, one has objects being propositions and morphisms being (classes) of proofs of one proposition from another.  The paper has a pretty extensive list of analogs between these domains, so go ahead and look in there for more!

Peter Selinger gave a talk about “Higher-Order Quantum Computation”.  This had to do with interesting phenomena that show up when dealing with “higher-order types” in quantum computers.  These are “data types”, as I just described – the “higher-order” types can be interpreted by blurring the distinction between a “system” and a “process”.  A data type describing a sytem we might act on might be $A$ or $B$.  A higher order type like $A \multimap B$ describes a process which takes something of type $A$ and returns something of type $B$.  One could interpret this as a black box – and performing processes on a type $A \multimap B$ is like studying that black box as a system itself.  This type is like an “internal hom” – and so one might like to say, “well, it’s dual to tensor – so it amounts to taking $A^* \otimes B$, since we’re in the category of Hilbert spaces”.  The trouble is, for physical computation, we’re not quite in the category where that works.  Because not all operators are significant: only some class of totally positive operators are physical.  So we don’t have the hom-tensor duality to use (equivalently, don’t have a well-behaved dual), and these types have to be considered in their own right.  And, because computations might not halt, operations studying a black box might not halt.  So in particular, a “co-co-qubit” isn’t the same as a qubit.  A co-qubit is a black box which eats a qubit and terminates with some halting probability.  A co-co-qubit eats a co-qubit and does the same.  If not for the halting probability, one could equally well see a qubit “eating” a co-co-qubit as the reverse.  But in fact they’re different.  A key fact in Peter’s talk is that quantum computation has new logical phenomena happening with types of every higher order.  Quantifying this (an open problem, apparently) would involve finding some equivalent of Bell inequalities that apply to every higher order of type.  It’s interesting to see how different quantum computing is, in not-so-obvious ways, from the classical kind.

Manoush Sadrzadeh gave a talk describing how “string diagrams” from monoidal categories, and representations of them, have been used in linguistics.  The idea is that the grammatical structure of a sentence can be build by “composing” structures associated to words – for example, a verb can be composed on left and right with subject and object to build a phrase.  She described some of the syntactic analysis that went into coming up with such a formalism.  But the interesting bit was to compare putting semantics on that syntax to taking a representation.  In particular, she described the notion of a semantic space in linguistics: this is a large-dimensional vector space that compares the meanings of words.  A rough but surprisingly effective way to clump words together by meaning just uses the statistics on a big sample of text, measuring how often they co-occur in the same context. Then there is a functor that “adds semantics” by mapping a category of string diagrams representing the syntax of sentences into one of vector spaces like this.  Applying the kind of categorical analysis usually used in logic to natural language seemed like a pretty neat idea – though it’s clear one has to make many more simplifying assumptions.

On the whole, it was a great conference with a great many interesting people to talk to – as you might guess from the fact that it took me three posts to comment on everything I wanted.

So as I mentioned in my previous post, I attended 80% of the conference “Categories, Quanta, Concepts”, hosted by the Perimeter Institute.  Videos of many of the talks are online, but on the assumption that not everyone will watch them all, I’ll comment anyway… 😉

It dealt with various takes on the uses of category theory in fundamental physics, and quantum physics particularly. One basic theme is that the language of categories can organize and clarify the concepts that show up here. Since there doesn’t seem to be a really universal agreement on what “fundamental” physics is, or what the concepts involved might be, this is probably a good thing.

There were a lot of talks, so I’ll split this into a couple of posts – this first one dealing with two obvious category-related themes – monoidal categories and toposes.  The next post will cover most of the others – roughly, focused on fundamentals of quantum mechanics, and on categories for logic and language.

Monoidal Categories

So a large contingent came from Oxford’s Comlab, many of them looking at ideas that I first saw popularized by Abramsky and Coecke about describing the features of quantum mechanics that appear in any dagger-compact category. This yields a “string diagram” notation for quantum systems. (An explanation of this system is given by Abramsky and Coecke – http://arxiv.org/abs/0808.1023 – or more concisely by Coecke – http://arxiv.org/abs/quant-ph/0510032).

Samson Abramsky talked about diagonal arguments. This is a broad class of arguments including Cantor’s theorem (that the real line is uncountable), Russell’s paradox in set theory (about the “set” of non-self-membered sets), Godel’s incompleteness theorem, and others. Abramsky’s talk was based on Bill Lawvere’s analysis of these arguments in general cartesian closed categories (CCC’s). The relevance to quantum theory has to do with “no-cloning” theorems – that quantum states can’t be duplicated. Diagonal arguments involve two capabilitiess: the ability to duplicate objects, and the ability to represent predicates (think of Godel numbering, for instance) which is related to a fixed point property. Generalizing to other monoidal categories, one still has representability: linear functionals on Hilbert spaces can be represented by vectors. But diagonal arguments fail since there is no diagonal $\Delta : H \rightarrow H \otimes H$.

Bob Coecke and Ross Duncan both spoke about “complementary observables”. Part of this comes from their notion of an “observable structure”, or “classical structure” for a quantum system. The intuition here is that this is some collection of observables which we can simultaneously observe, and such that, if we restrict to those observables, and states which are eigenstates for them, we can treat the whole system as if it were classical. In particular, this gives us “copy” and “destroy” operations for states – these maps and their duals actually turn out to define a Frobenius algebra. In finite-dimensional Hilbert spaces, this is equivalent to choosing an orthonormal basis.

Complementary observables is related to the concept of mutually unbiased bases. So the bases $\{v_i\}$ and $\{w_j\}$ are unbiased if all the inner products $\langle v_i , w_j \rangle$ have the same magnitude. If these bases are associated to observables (say, they form a basis of eigenvectors), then knowing a classical value of one observable gives no information about the other – all eigenstates are equally likely. For a visual image, think of two sets of bases for the plane, rotated 45 degrees relative to each other. Each basis vector in one has a projection of equal length onto both basis vectors of the other.

Thinking of the orthonormal bases as “observable structures”, the mutually unbiased ones correspond to “complementary” observables: a state which is classical for one observable (i.e. is an eigenstate for that operator) is unbiased (i.e. has equal probablities of having any value) for the other observable. Labelling the different structures with colours (red and green, usually), they could diagrammatically represent states being classical or unbiased in particular systems.

This is where “phase groups” come into play. The setup is that we’re given some system – the toy model they often referred to was a spinning particle in 3D – and an observable system (say, just containing the observable “spin in the X direction”). Then there’s a group of symmetries of the system which leave that observable untouched (in that example, the symmetries are rotation about the X axis). This is the “phase group” for that observable.

Bill Edwards talked about phase groups and how they can be used to classify systems. He gave an example of a couple of toy models with six states each. One was based on spin (the six states describe spins about each axis in 3-space in each direction). The other, due to Robert Spekkens, is a “hidden variable” theory, where there are four possible “ontic” states (the “hidden” variable), but the six “epistemic” states only register whether the state lies in of six possible PAIRS of ontic states. The two toy models resemble each other at the level of states, but the phase groups are different: the truly “quantum” one has a cyclic group $\mathbb{Z}_4$ (for the X-spin observable, it’s generated by a right-angled rotation about the X axis); the “hidden variable” model, which has some quantum-mechanics-like features, but not all, has phase group $\mathbb{Z}_2 \times \mathbb{Z}_2$. The suggestion of the talk was that this phase group distinguishes “local” from “nonlocal” systems (i.e. ones with hidden variable models and ones without).

Marni Sheppard also gave a talk about Mutually Unbiased Bases, p-adic arithmetic, and algebraic geometry over finite fields, which I find hard to summarize because I don’t understand all those fields very well. Roughly, her talk made a link between quantum mechanics and an axiomatic version of projective geometry (Hilbert spaces in QM ought to be projective, after all, so this makes sense).  There was also a connection between mutually unbiased bases and finite fields, but again, this sort of escaped me.

Also in this group was Jamie Vicary, whom I’ve been working with on a project about the categorified harmonic oscillator.  His talk, however, was about n-Hilbert spaces, and n-categorical extended TQFT.  The basic point is that a TQFT assigns a number to a closed n-manifold, and a Hilbert space to each (n-1)-manifold (such as a boundary between two parts of a closed one), and if the TQFT is fully local (i.e. can be derived from, say, a triangulation), this can be continued to have it assign k-Hilbert spaces to (n-k)-manifolds for all k up to n.  He described the structure of 2-Hilbert spaces, and also monoidal ones (as many interesting cases are), and how they can all be realized (in finite dimensions, at least) as categories of representations of supergroupoids.  Part of the point of this talk was to suggest how not just dagger-compact categories, but general n-categories should be useful for quantum theory.

Toposes

The monoidal category setting is popular for dealing with quantum theories, since it abstracts some properties of Hilbert spaces, which they’re usually modelled in.  Topos theory is usually thought of as a generalization of the category of sets, and in particular they model intuitionistic classical, not quantum, logic.  So the talk by Andreas Döring (based on work with Christopher Isham – see many of Andreas’ recent papers) called “Why Topos Theory in the Foundations of Physics?” is surprising if you haven’t heard this idea before.  One motivation could be described in terms of the Kochen-Specker theorem, which, roughly, says that a quantum theory – involving observables which are operators on a Hilbert space of dimension at least three – can’t be modeled by a “state space”.  That is, it’s not the case that you can simultaneously give definite values to all the observables in a consistent way – in ANY state!  (That is, it’s not just the generic state: there is no state at all which corresponds to the classical picture of a “point” in some space parametrized by the observables.)

Now, part of the point is that there’s no “state space” in the category of sets – but maybe there is in some other topos!  And sure enough, the equivalent of a state space turns out to be a thing they call the “spectral presheaf” for the theory.  It’s an object in some topos.  The KS theorem becomes a statement that it has no “global points”.  To see what this means, you have to know what the spectral presheaf is.

This is based on the assumption that one has a (noncommutative) von Neumann algebra of operators on a Hilbert space – among them, the observables we might be interested in.  The structure of this algebra is supposed to describe some system.  Now you might want to look for subalgebras of it which are abelian.  Why?  Because a system of commuting operators, should they be observables, are ones which we CAN assign values to simultaneously – there’s no issue of which order we do measurements in.  Call this a “context” – a choice of subalgebra making the system look classical.  So maybe we can describe a “state space” in a context: so what?

Well, the collection of all such contexts forms a poset – in fact, lattice – in fact, a complete Heyting algebra.  These objects are just the same (object-wise) as “locales” (a generalization from topological spaces, and their lattice of open sets).  The topos in question is the category of presheaves on this locale, which is to say, of contravariant functors to Set.  Which is to say… a way of assigning a set (the “state space” I mentioned), with a way of restricting sets along inclusion maps.  This restriction can be a bit rough (in fact, the fact that restriction can be quite approximate is just where uncertainty principles and the like come from).  The main point is that this “spectral presheaf” (the assignment of local state spaces to each context) supports a concept of logic, for reasoning about the system it describes.  It’s a lot like the logic of sets, but operations happen “context-by-context”.  A proposition has a truth value which is a “downset” in the lattice of contexts – the collection of contexts where the proposition is true.  A proposition just amounts to a subobject of the spectral presheaf by what they call “daseinization” – it’s the equivalent of a proposition being a subset of a configuration space (where the statement is true).

One could say a lot more, but this is a blog post, after all.

There are philosophical issues that this subject seems to provoke – the sign of an interesting theory is that it gets people arguing, I suppose.  One is the characterization of this as a “neo-realist interpretation” of quantum theory.  A “naive realist” interpretation would be one that says a “state” is just a way of saying what all the values of all the observable quantities is – to put it another way, of giving definite truth values to all definite “yes/no” questions.  This is just what the KS theorem says can’t happen.  The spectral presheaf is supposedly “neo-realist” because it does almost these things, but in an exotic topos (of presheaves on the locale of all classical contexts).  As you might expect, this is a bit of a head-scratcher.

I spent most of last week attending four of the five days of the workshop “Categories, Quanta, Concepts”, at the Perimeter Institute.  In the next few days I plan to write up many of the talks, but it was quite a lot.  For the moment, I’d like to do a little writeup on the talk I gave.  I wasn’t originally expecting to speak, but the organizers wanted the grad students and postdocs who weren’t talking in the scheduled sessions to give little talks.  So I gave a short version of this one which I gave in Ottawa but as a blackboard talk, so I have no slides for it.

Now, the workshop had about ten people from Oxford’s Comlab visiting, including Samson Abramsky and Bob Coecke, Marni Sheppard, Jamie Vicary, and about half a dozen others.  Many folks in this group work in the context of dagger compact categories, which is a nice abstract setting that captures a lot of the features of the category $Hilb$ which are relevant to quantum mechanics.  Jamie Vicary had, earlier that day, given a talk about n-dimensional TQFT’s and n-categories – specifically, n-Hilbert spaces.  I’ll write up their talks in a later,  but it was a nice context in which to give the talk.

The point of this talk is to describe, briefly, $Span(Gpd)$ – as a category and as a 2-category; to explain why it’s a good conceptual setting for quantum theory; and to show how it bridges the gap between Hilbert spaces and 2-Hilbert spaces.

History and Symmetry

In the course of an afternoon discussion session, we were talking about the various approaches people are taking in fundamentals of quantum theory, and in trying to find a “quantum theory of gravity” (whatever that ends up meaning).  I raised a question about robust ideas: basically, it seems to me that if an idea shows up across many different domains, that’s probably a sign it belongs in a good theory.  I was hoping people knew of a number of these notions, because there are really only two I’ve seen in this light, and really there probably should be more.

The two physical  notions that motivate everything here are (1) symmetry, and (2) emphasis on histories.  Both ideas are applied to states: states have symmetries; histories link starting states to ending states.  Combining them suggests histories should have symmetries of their own, which ought to get along with the symmetries of the states they begin and end with.

Both concepts are rather fundamental. Hermann Weyl wrote a whole book, “Symmetry”, about the first, and wrote: As far as I can see, all a-priori statements in physics are based on symmetry. From diffeomorphism invariance in general relativity, to gauge symmetry in quantum field theory, to symmetric tensor products involved in Fock space, through classical examples like Noether’s theorem. Noether’s theorem is also about histories: it applies when a symmetry holds along an entire history of a system: in fact, Langrangian mechanics generally is all about histories, and how they’re selected to be “real” in a classical system (by having a critical value of the action functional). The Lagrangian point of view appears in quantum theory (and this was what Richard Feynman did in his thesis) as the famous “sum over histories”, or path integral. General relativity embraces histories as real – they’re spacetimes, which is what GR is all about. So these concepts seem to hold up rather well across different contexts.

I began by drawing this table:

 $Sets$ $Span(Sets) \rightarrow Rel$ $Grpd$ $Span(Grpd)$

The names are all those of categories. Moving left to right moves from a category describing collections of states, to one describing states-and-histories. It so happens that it also takes a cartesian category (or 2-category) to a symmetric monoidal one. Moving from top to bottom goes from a setting with no symmetry to one with symmetry. In both cases, the key concept is naturally expressed with a category, and shows up in morphisms. Now, since groupoids are already categories, both of the bottom entries properly ought to be 2-categories, but when we choose to, we can ignore that fact.

Why Spans?

I’ve written a bunch on spans here before, but to recap, a span in a category $C$ is a diagram like: $X \stackrel{s}{\leftarrow} H \stackrel{t}{\rightarrow} Y$. Say we’re in $Sets$, so all these objects are sets: we interpret $X$ and $Y$ as sets of states. Each one describes some system by collecting all its possible (“pure”) states. (To be better, we could start with a different base category – symplectic manifolds, say – and see if the rest of the analysis goes through). For now, we just realize that $H$ is a set of histories leading the system $X$ to the system $Y$ (notice there’s no assumption the system is the same). The maps $s,t$ are source and target maps: they specify the unique state where a history $h \in H$ starts and where it ends.

If $C$ has pullbacks (or at least any we may need), we can use them to compose spans:

$X \stackrel{s_1}{\leftarrow} H_1 \stackrel{t_1}{\rightarrow} Y \stackrel{s_2}{\leftarrow} H_2 \stackrel{t_2}{\rightarrow} Z \stackrel{\circ}{\Longrightarrow} X \stackrel{S}{\leftarrow} H_1 \times_Y H_2 \stackrel{T}{\rightarrow} Z$

The pullback $H_1 \times_Y H_2$ – a fibred product if we’re in $Sets$ – picks out pairs of histories in $H_1 \times H_2$ which match at $Y$. This should be exactly the possible histories taking $X$ to $Z$.

I’ve included an arrow to the category $Rel$: this is the category whose objects are sets, and whose morphisms are relations. A number of people at CQC mentioned $Rel$ as an example of a monoidal category which supports toy models having some but not all features of quantum mechanics. It happens to be a quotient of $Span(Sets)$. A relation is an equivalence class of spans, where we only notice whether the set of histories connecting $x \in X$ to $y \in Y$ is empty or not. $Span(Sets)$ is more like quantum mechanics, because its composition is just like matrix multiplication: counting the number of histories from $x$ to $y$ turns the span into a $|X| \times |Y|$ matrix – so we can think of $X$ and $Y$ as being like vector spaces.

In fact, there’s a map $L : Span(Sets) \rightarrow Hilb$ taking an object $X$ to $\mathbb{C}^X$ and a span to the matrix I just mentioned, which faithfully represents $Span(Sets)$. A more conceptual way to say this is: a function $f : X \rightarrow \mathbb{C}$ can be transported across the span. It lifts to $H$ as $f \circ s : H \rightarrow \mathbb{C}$. Getting down the other leg, we add all the contributions of each history ending at a given $y$: $t_*(s \circ f) = \sum_{t(h)=y} f \circ s (h)$.

This “sum over histories” is what matrix multiplication actually is.

Why Groupoids?

The point of groupoids is that they represent sets with a notion of (local) symmetry. A groupoid is a category with invertible morphisms. Each such isomorphism tells us that two states are in some sense “the same”. The beginning example is the “action groupoid” that comes from a group $G$ acting on a set $X$, which we call $X /\!\!/ G$ (or the “weak quotient” of $X$ by $G$).

This suggests how groupoids come into the physical picture – the intuition is that $X$ is the set (or, in later variations, space) of states, and $G$ is a group of symmetries.  For example, $G$ could be a group of coordinate transformations: states which can be transformed into each other by a rotation, say, are formally but not physically different.  The Extended TQFT example comes from the case where $X$ is a set of connections, and $G$ the group of gauge transformations.  Of course, not all physically interesting cases come from a single group action: for the harmonic oscillator, the states (“pure states”) are just energy levels – nonnegative integers.  On each state $n$, there is an action of the permutation group $S_n$ – a “local” symmetry.

One nice thing about groupoids is that one often really only wants to think about them up to equivalence – as a result, it becomes a matter of convention whether formally different but physically indistinguishable states are really considered different.  There’s a side effect, though: $Gpd$ is a 2-category.  In particular, this has two consequences for $Span(Gpd)$: it ought to have 2-morphisms, so we stop thinking about spans up to isomorphism.  Instead, we allow spans of span maps as 2-morphisms.  Also, when composing spans (which are no longer taken up to isomorphism) we have to use a weak pullback, not an ordinary one.  I didn’t have time to say much about the 2-morphism level in the CQC talk, but the slides above do.

In any case, moving into $Span(Gpd)$ means that the arrows in the spans are now functors – in particular, a symmetry of a history$h$  now has to map to a symmetry of the start and end states, $s(h)$ and $t(h)$.  In particular, the functors give homomorphisms of the symmetry groups of each object.

Physics in Hilb and 2Hilb

So the point of the above is really to motivate the claim that there’s a clear physical meaning to groupoids (states and symmetries), and spans of them (putting histories on an even footing with states).  There’s less obvious physical meaning to the usual setting of quantum theory, the category $Hilb$ – but it’s a slightly nicer category than $Span(Gpd)$.  For one thing, there is a concept of a “dual” of a span – it’s the same span, with the roles of $s$ and $t$ interchanged.  However (as Jamie Vicary pointed out to me), it’s not an “adjoint” in $Span(Gpd)$ in the technical sense.  In particular, $Span(Gpd)$ is a symmetric monoidal category, like $Hilb$, but it’s not “dagger compact”, the kind of category all the folks from Oxford like so much.

Now, groupoidification lets us generalize the map $L : Span(Sets) \rightarrow Hilb$ to groupoids making as few changes as possible.  We still use Hilbert space $\mathbb{C}^X$, but now $X$ is the set of isomorphism classes of objects in the groupoid.  The “sum over histories” – in other words, the linear map associated to a span – is found in almost the same way, but histories now have “weights” found using groupoid cardinality (see any of the papers on groupoidification, or my slides above, for the details).  This reproduces a lot of known physics (see my paper on the harmonic oscillator; TQFT’s can also be defined this way).

While this is “as much like” linearization of $Span(Set)$ as possible in some sense, it’s not exactly analogous.  It also is rather violent to the structure of the groupoids: at the level of objects it treats $X /\!\!/ G$ as $X/G$. At the morphism level, it ignores everything about the structure of symmetries in the system except how many of them there are.   Since a groupoid is a category, the more direct analogy for $\mathbb{C}^X$ – the set of functions (fancier versions use, say, $L^2$ functions only) from $X$ to $\mathbb{C}$ is $Hilb^G$ – the category of functors from a groupoid into $Hilb$.  That is, representations of $X$.

One of the attractions here is that, because of a generalization of Tanaka-Krein duality, this category will actually be enough to reconstruct the groupoid if it’s reasonably nice.  The representation of $Span(Gpd)$ in $2Hilb$, unlike in $Hilb$ is actually faithful for objects, at least for compact or finite groupoids.

Then you can “pull and push” a representation$F$ across a span to get $t_*(F \circ s)$ – using $t_*$, the adjoint functor to pulling back.  This is the 1-morphism level of the 2-functor I call $\Lambda$, generalizing the functor $L$ in the world of sets.  The result is still a “direct sum over histories” – but because we’re dealing with pushing representations through homomorphisms, this adjoint is a bit more complicated than in the 0-category world of $\mathbb{C}$.  (See my slides or paper for the details).  But it remains true that the weights and so forth used in ordinary groupoidification show up here at the level of 2-morphisms.  So the representation in $2Hilb$ is not a faithful representation of the (intuitively meaningful) category $Span(Gpd)$ either.  But it does capture a fair bit more than Hilbert spaces.

One point of my talk was to try to motivate the use of 2-Hilbert spaces in physics from an a-priori point of view.  One thing I think is nice, for this purpose, is to see how our physical intuitions motivate $Span(Gpd)$ – a nice point itself – and then observe that there is this “higher level” span around:

$Hilb \stackrel{|\cdot |}{\leftarrow} Span(Gpd) \stackrel{\Lambda}{\rightarrow} 2Hilb$

Further Thoughts

Where can one take this?  There seem to be theories whose states and symmetries naturally want to form n-groupoids: in “higher gauge theory“, a sort of  gauge theory for categorical groups, one would have connections as states, gauge transformations as symmetries, and some kind of  “symmetry of symmetries”, rather as 2-categories have functors, natural transformations between them, and modifications of these.  Perhaps these could be organized into n-dimensional spans-of-spans-of-spans… of n-groupoids.  Then representations of an n-groupoid – namely, n-functors into $(n-1)-Hilb$ – could be subjected to the kind of “pull-push” process we’ve just looked at.

Finally, part of the point here was to see how some fundamental physical notions – symmetry and histories – appear across physics, and lead to $Span(Gpd)$.  Presumably these two aren’t enough.  The next principle that looks appealing – because it appears across domains – is some form of an action principle.

But that would be a different talk altogether.