## January 4, 2008

### Dijkgraaf-Witten and its Categorification by Martins and Porter

#### Posted by Urs Schreiber

I was looking at

J. F. Martins and T. Porter
On Yetter’s invariants and an extension of the Dijkgraaf-Witten invariant to categorical groups
(tac)

which João Martins pointed me to in a comment to the entry BF-Theory as a higher gauge theory. (John discussed this paper here a while back).

On the train back home, this inspired me to write the following notes, which happen to be mostly about ordinary Dijkgraaf-Witten theory, but try to put the general context into perspective.

[Update: Typed notes on this topic are now here: On $\Sigma$-models and nonabelian differential cohomology]

The authors categorify Dijkgraaf-Witten theory, which is a sigma-model theory with target space a one-object groupoid, to a sigma-model theory with target space a strict one-object 2-groupoid.

Before talking about what these authors do, I recall and extend what we have said here before about the $n$-categorical sigma-model formulation of Dijkgraaf-Witten theory – which is not something these authors consider, but something I enjoy translating everything into, because, in particular, it explains the curious “triangulation weights” that appear in this context.

Recall (for instance from Canonical measures on configuration spaces) Dijkgraaf-Witten theory is the sigma-model type quantum field theory whose

-parameter spaces are of the form $P_1(X)$

for $X$ some 3-manifold with chosen triangulation and $P_1(X)$ the groupoid generated from the edges of the triangulation modulo the faces (hence a finite versoin of the fundamental 1-groupoid of $X$);

- target space is the groupoid $\mathbf{B}G := \{\bullet \stackrel{g}{\to} \bullet | g \in G\}$ whose geometric realization yields the classifying space $|\mathbf{B}G| \simeq B G$ of some finite group $G$;

- background field is a parallel transport pseudofunctor $\mathrm{tra} : \mathbf{B}G \to \mathbf{B}^3 U(1) \,,$ where $\mathbf{B}^3 U(1)$ is the strict 3-groupoid with whose space of 3-morphisms is $U(1)$, which is entirely determined by its associator, which in turn encodes precisely a group 3-cocycle on $G$;

- configuration space is the internal hom $\mathrm{conf} = \mathrm{hom}_{\mathrm{Cat}}(P_1(X), \mathbf{B} G)$ wich is the groupoid whose objects are flat $G$-connections on $X$ and whose morphisms are gauge transformations of these.

Therefore the action of this theory is the trangression of the background field to the configuration space $S = \mathrm{tg} \mathrm{tra} := \mathrm{hom}(P_1(X),\mathrm{tra}) : \mathrm{hom}(P_1(X),\mathbf{B}G) \to \mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim \,,$ where $/\sim$ means that we form the set of equivalence classes of objects.

This is the first point where something interesting happens. Let’s figure out what $\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim$ is.

Objects of $\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))$ are pseudofunctors from $P_1(X)$ to $\mathbf{B}^3 U(1)$ (weak 3-functors if you wish, with $P_1(X)$ regarded as a 3-category). The only nontrivial aspect of these pseudofunctors are their associators and hence, I think, they are in bijection with colorings of the tetrahedra in $X$ by elements in $U(1)$. Moreover, the morphisms between such pseudofunctors are in bijection with colorings of triangles in $X$ with elements in $U(1)$, where from a coloring of tetrahedra and and one of triangles we get to a new coloring of tetrahedra by multiplying each tetrahedron label by the labels on its faces, with orientations taken care of suitably.

What labels equivalence classes of objects in $P_1(X) \to \mathbf{B}^3 U(1)$? If two objects are connected by a morphism, then their “integrated $U(1)$-label” $\prod_{tetrahedra t in X} (label of tetrahedron t)$ is the same.

The converse does not quite hold in general, unless $X$ is sufficiently nice, I think. But if $X$ is the 3-sphere, we can essentially model it by $\mathbf{B}^3 \mathbb{Z}$ and then we’d have $\mathrm{hom}(\mathbf{B}^3\mathbb{Z}, \mathbf{B}^3 U(1))/\sim = U(1) \,.$

So let me assume that $X$ is the 3-sphere from now on. Then we have found that the action of Dijkgraaf-Witten theory, which was the transgression of the background field on target space $\mathbf{B}G$ to configuration space, is the map $S := \mathrm{hom}(P_1(X),\mathrm{tra}) : \mathrm{hom}(P_1(X),\mathbf{B}G) \to \mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim$ which sends each flat $G$-connection $A$ on $X$ to the number $S : A \mapsto \prod_{tetrahedra t in X} \mathrm{tra}(A(t)) \in U(1) \,.$

This action we want to integrate over all of configuration space. To do so we should regard it as a functor on configuration space and somehow for its colimit over all of configuration space. The best current idea we have for what that actually means is:

we sum the action over all objects of configuration space and weigh each contribution by the Leinster measure $d\mu$ of configuration space, which here amounts just to weighing it by the groupoid cardinality

$d\mu : A \mapsto (number of morphisms starting at A) \,.$

We get one morphism starting at any flat $G$-connection on $X$ for every choice of element in $G$ per vertex of $X$. If there are $v$ many vertices, this means that $d\mu(A) = \frac{1}{|G|^v} \,.$

So, finally, the partition function of Dijkgraaf-Witten theory for given parameter space $X$, which we want to think of as $Z(X) = colim_{hom(P_1(X),\mathbf{B}G)} \mathrm{tg}(\mathrm{tra})$

is

$Z(X) = \sum_{flat G-connections A} \prod_{tetrahedra t in X} \mathrm{tra}(A(t)) \; \frac{1}{|G|^v} \,.$

Unfortunately, despite lots of thinking and conjecturing and toy-example-checking here on the $n$-Café it is still not quite clear how the morally right way to think of a colimit here makes strictly sense as a colimit, and so for the time being we have to invoke by hand the rule that “path integral means sum of transgressed background field weighted by the Leinster measure”. But still, it’s pretty cool.

(I should add that the picture which I am drawing here has been painted a lot by Simon Willerton and Bruce Bartlett, based on old work by Quillen. What I am adding are some, I think, refinements concerning aspects like the background field, parallel transport and transgression.)

Okay, that sets the stage. This is Dijkgraaf-Witten theory. One shows that this $Z$ is a homotopy invariant of 3-manifolds (and hence in particular independent of the chosen triangulation). In order for this to be true, the precise nature of the Leinster measure $d\mu(A) = \frac{1}{|G|^v}$ is crucial.

Now we want to know: what happens to this setup if target space is not necessarily a (finite) 1-group $\mathbf{B}G$ but an arbitrary strict (finite) 2-group $\mathbf{B}G_{(2)}$ where $G_{(2)}$ comes from the crossed module $(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(G)) \,.$

This is best approached in two steps:

- first using just the trivial background field, concentrating on what happens to the path integral domain and Leinster measure;

- then introducing also nontrivial background fields.

The first step was done by D. Yetter in his TQFTs from homotopy 2-types. There he finds what we’d call the “Leinster 2-measure” for the path integral, namely the combinatorial weight $d\mu$ that depends on the chosen triangulation, by fixing some weighting, and then demonstrating that it yields a partition function $Z$ which is a homotopy invariant.

Martins and Porter then also include a nontrivial background field in this context.

They also generalize the proof that the partition function $Z$ is a homotopy invariant to higher dimensional parameter spaces. For $n=4$ and trivial background field, this is also done by Girelli-Pfeiffer-Popescu.

Posted at January 4, 2008 10:38 PM UTC

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## 8 Comments & 3 Trackbacks

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

Proposition. The combinatorial weighting for the “categorified DW” theory introduced by Yetter (and studied by Girelli, Pfeiffer, Popescu, Porter, Martins etc.) is indeed the Leinster measure of the corresponding configuration space.

Recall that for ordinary DW theory for finite group $G$ config space is the groupoid $\mathrm{hom}_{Cat}(P_1(X), \mathbf{B}G)$ and that since there are $|G|^\nu$ many ways to assign group element to the $\nu$ vertices of $X$, there are $|G|^\nu$ many natural transformations starting at any functor $P_1(X) \to \mathbf{B}G$ and therefore the Leinster measure of the hom-groupoid is $d\mu = \frac{1}{|G|^\nu} \,.$

Now, Yetter et al find that when target space is replaced by $\mathbf{B}G_{(2)}$ for $G_{(2)}$ the finite strict 2-group coming from the strict crossed module $H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H) \,,$ then the corresponding combinatorial weight becomes $|G|^{- \nu} |H|^{-\nu + \lambda} \,,$ where $\nu$ is the number of vertices, as before, and where $\lambda$ is the number of edges in $X$.

(See for instance theorem 2.13 on p. 127 of Martins-Porter’s On Yetter’s invariants or the discussion below theorem III.2 on p. 6 of Girelli-Pfeiffer-Popescu’s BF to BFCG (while the factor is not manifest in their expression (15))).

I am claiming that this factor is precisely the Leinster measure on the configuration space

$conf = \mathrm{hom}_{2Cat}(P_1(X), \mathbf{B}G_{(2)})/\sim$

Here I am taking, to replace awkward pseudofunctors with nicer strict 2-functors, $P_1(X)$ now to be the strict 2-groupoid generated from the triangles in $X$ (which is a manifold with chosen triangulation, recall).

Then the above hom-groupoid is supposed to be the 1-groupoid obtained by passing to equivalence classes of 1-morphisms in the 2-groupoid whose objects are strict 2-functors $P_1(X) \to \mathbf{B}G_{(2)}$, whose morphisms are pseudonatural transformations of these, and whose 2-morphisms are modifications of those.

To prove this, we need to compute how many 1-morphisms in $conf$ there start at any given object.

But that’s easy: a 1-morphism $g : A \to A'$ in the 2-groupoid of strict 2-functors $P_1(X) \to \mathbf{B}G_{(2)}$ is a transformation whose component is an assignment of squares in $\mathbf{B}G_{(2)}$ to edges in $X$:

$g : (x \stackrel{\gamma}{\to} y) \;\;\;\; \mapsto \;\;\;\; \array{ \bullet &\stackrel{A(\gamma)}{\to}& \bullet \\ \downarrow^{g(x)} & \Downarrow^{g(\gamma)} & \downarrow^{g(y)} \\ \bullet &\stackrel{A'(\gamma)}{\to}& \bullet } \,.$

By the general Yoga of strict 2-groups, we know that the square on the right is entirely and precisely fixed by

- the source 2-functor value $A(\gamma) \in G$ on the top;

- the two transformation elements $g(x), g(y) \in G$ on the two sides;

- and the label $g(\gamma) \in H$ of the 2-morphism filling the square.

This uniquely determines the value $A'(\gamma)$ of the pseudofunctor which is the target of the transformation whose component $g$ is.

Therefore, clearly, there are precisely

$|G|^\nu |H|^\lambda$

many 1-morphisms in $\mathrm{hom}_{2Cat}(P_1(X), \mathbf{B}G_{(2)})$

starting at any given object.

But some of them are related by 2-morphisms

To figure out many, we need to compute how many modifications $\eta : g \to g'$ there are starting at any transformation $g$.

But again the same logic applies: such a transformations is given by a component map which sends vertices in $X$ to 2-cells in $\mathbf{B}G_{(2)}$:

$\eta : x \mapsto \array{ & \nearrow \searrow^{g(x)} \\ \bullet & \Downarrow^{\eta(x)}& \bullet \\ & \searrow \nearrow_{g'(x)} }$ for $\eta(x) \in H$.

Again one sees, by drawing the relevant naturality tin-can diagram which I won’t bother to do here in MathML, that for each such choice of assignments of elements in $H$ to vertices in $X$ there is precisely one modification $\eta : g \to g'$.

This means that we have

- $|G|^\nu |H|^\lambda$ 1-morphisms emanating at each object

and

- $|H|^\nu$ 2-morphisms emanating at every morphism.

The Leinster measure in this situation is $d\mu(A) = |G|^{-\nu} |H|^{\nu - \lambda} \,,$ precisely the combinatorial factor which people found makes the Yetter state sum model a topological invariant.

[Update: A more detailed discussion of this is now in On $\Sigma$-models and nonabelian differential cohomology.]

Posted by: Urs Schreiber on January 5, 2008 3:13 PM | Permalink | Reply to this

### Season of the “wich”; Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

“… wich is the groupoid whose objects are flat G-connections” is more likely to be “which” than “witch.”

And I’ve asked before, but didn’t understand the answer, does it matter that not every polyhedron can be tetrahedralized?

Posted by: Jonathan Vos Post on January 5, 2008 11:28 PM | Permalink | Reply to this

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

Dear Urs,

very nice post!

Your argument to identify the (mysterious) factor that makes Yetter’s state sum a topological invariant, for any simplicial complex, seems similar to the one we gave in our paper (see pages 134 and 135), basically counting the number of crossed module 1-fold and 2-fold homotopies starting in a certain crossed module map $P_2(X)\to G_2$.

It would be interesting to verify how your proof translates to the equivalent? language of morphisms of crossed modules and (1- and 2-fold) homotopies between them. This is the language we used in our paper, so that we could appeal to the very strong framework of simplicial homotopy theory (and classifying spaces of crossed complexes), and identify all the terms with known homotopy invariants.

Posted by: João Martins on January 7, 2008 4:12 PM | Permalink | Reply to this

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

Hi João,

thanks for your message!

It would be interesting to verify how your proof translates to the equivalent? language of morphisms of crossed modules and (1- and 2-fold) homotopies between them.

I need to have another look at your paper, am very busy right this moment. But it would surprise me if the crossed module homotopies you are referring to aren’t precisely what one gets by interpreting crossed modules as strict one-object 2-groupoids and then looking at pseudonatural transformations and their modifications of 2-functors with values in these. That’s what I did phrase my argument in.

Posted by: Urs Schreiber on January 7, 2008 6:49 PM | Permalink | Reply to this

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

João,

I have a bit of time now to get back to this.

This is very interesting. I didn’t notice your lemmas 2.27 and 2.28 before, which clearly must be completely analogous to the 2-functorial argument I gave above.

Unfortunately I haven’t learned about “crossed module homotopies” before and will have to do a bit of literature search.

I gather I should start with Brown & Icen’s Homotopies and automorphisms of crossed modules of groupoids. Hm, I see that Ronnie Brown talks about that in a reply appended at the end of John’s week 223!

Surely there must be some well known result that relates crossed modules homotopies to higher morphisms between some $n$-groupoidal structures (possibly $n$-fold-groupoidal, I suppose)?

Posted by: Urs Schreiber on January 7, 2008 10:13 PM | Permalink | Reply to this

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

I would like to make a general comment on models for higher groupoids. A slew of these are described briefly in
`Groupoids and crossed objects in algebraic topology’, Homology, homotopy and applications, 1 (1999) 1-78.
I have made use of crossed complexes and cubical omega-groupoids with connections. Crossed complexes were shown equivalent to globular omega groupoids in
(with P.J. HIGGINS), “The equivalence of $\infty$-groupoids and crossed complexes”, {\em Cah. Top. G'eom. Diff.} 22 (1981) 371-386.

A monoidal closed structure on crossed complexes is given explicitly in
(with P.J. HIGGINS), “Tensor products and homotopies for $\omega$-groupoids and crossed complexes”, {\em J. Pure Appl. Alg.} 47 (1987) 1-33.

This is one of the aspects that gives crossed complexes their power. The internal hom for crossed complexes involves of course higher homotopies.
I would find it difficult to do analogous things for the globular case.

I have been unable to DO anything much with the globular omega-groupoids, either calculate, relate to chain complexes, of prove Higher Homotopy van Kampen Theorems directly, so I just prefer the other models!

Another aspect of crossed complexes is the utility of free crossed complexes. So we have useful notions of free crossed resolutions of groups, and some results on calculating them. They are useful in Schreier theory (Tim and I have a paper on that.) See also http://arxiv.org/abs/0708.1677
and
http://www.bangor.ac.uk/r.brown/nonab-a-t.html
The last links to a recently found 2 page document by Grothendieck on his early intentions for Pursuing Stacks.

Ronnie Brown

Posted by: Ronnie Brown on January 11, 2008 3:43 PM | Permalink | Reply to this

### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

I would be very surprised if the following hasn’t been noticed before, but in the light of the above discussion and given Ronnie Brown’s remark at the end of TWF 223

It looks as if it would be better expressed in terms of automorphisms of 2-groupoids: good marks to anyone who writes it down in that way!

I’ll say it nevertheless, just for the fun of it:

Proposition. For crossed modules of groups, the crossed module homotopies in definition 2.1 and proposition 2.2 of Brown & Icen correspond precisely to pseudonatural transformations of strict 2-functors between the strict one-object 2-groupoids canonically isomorphic to the crossed modules in question.

Proof.

Given a crossed module $G_{(2)} = ( H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H) )$ of groups, we obtain a strict one-object 2-groupoid $\mathbf{B} G_{(2)} = \left\lbrace \array{ & \nearrow \searrow^{g\;\;} \\ \bullet &\Downarrow^h& \bullet \\ & \searrow \nearrow_{g' = t(h)g} } \;\; | \;\; g,g' \in G, h \in H \right\rbrace$ whose vertical composition is given by the product in $H$ and whose horizontal composition is given by the product in $H \ltimes_\alpha G$.

This establishes a canonical isomorphism between crossed modules of groups and strict one-object 2-groupoids, as is well known.

Strict 2-functors between such 2-groupoids correspond precisely to the the morphisms of the corresponding crossed modules as on the bottom of p. 4 in Brown & Icen, as one directly sees.

Pseudonatural transformations $\array{ & \nearrow \searrow^{F_1} \\ \mathbf{B}G_{(2)} &\Downarrow^\eta& \mathbf{B}G'_{(2)} \\ & \searrow \nearrow_{F_2} }$ between such strict 2-functors are given by component functors $\eta : (\bullet \stackrel{g}{\to} \bullet) \;\; \mapsto \;\; \left( \array{ \bullet &\stackrel{F_1(g)}{\to}& \bullet \\ \downarrow^{\eta(\bullet)} &\Downarrow^{\eta(g)}& \downarrow^{\eta(\bullet)} \\ \bullet &\stackrel{F_2(g)}{\to}& \bullet } \right) \,,$ where the respected composition on the right is horizontal pasting of these squares in $\mathbf{B}G'_{(2)}$.

This horizontal pasting involves one rewhiskering, which is the twist in the third displayed equation on p.5 of Brown&Icen.

The mere existence of the square on the right, with the given labels, is the fifth equation on that page.

Finally, the above $\eta$ has to make the naturality tin can diagram 2-commute. That is the sixth equation on that page (evaluated only, which is sufficient, on 2-morphisms in $\mathbf{B}G_{(2)}$ whose source is the identity 1-morphism).

Posted by: Urs Schreiber on January 7, 2008 10:56 PM | Permalink | Reply to this
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### Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

Hi!

I forgot to say this at due time: for people interested in Yetter’s invariant and related stuff (not considering crossed module cohomology classes), two years ago I wrote this:

On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex

which is an article mainly on Algebraic Topology, but among other things it contains two different proofs of the existence of Yetter’s invariant for general CW-complexes, along with its geometric interpretation (in the general case of crossed complexes).

For people interested in knotted surfaces, there is also this:

The Fundamental Crossed Module of the Complement of a Knotted Surface

which as a byproduct, calculates Yetter’s invariant of the complement of a knotted surface in $S^4$.

Posted by: João Faria Martins on July 20, 2008 1:53 PM | Permalink | Reply to this
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