## June 24, 2008

### Tim Porter on Formal Homotopy Quantum Field Theories and 2-Groups

#### Posted by David Corfield

Guest post by Bruce Bartlett

I’d like to give something of a report-back on Tim Porter’s second talk at Barcelona, on Formal Homotopy Quantum Field Theories and 2-Groups (slides).

Firstly let me say that this was the first time I had the pleasure of meeting Tim. If anyone at the Café would like to know, Tim is one of those curious and charming breeds of Englishmen who was born in Wales but occasionally lapses into an Irish accent, and whose constitution requires for its good upkeep a steady diet of fine European cuisine (especially seafood), regular cups of rooibos tea, a daily dollop of French, bird sightings (you live in Buenos Aires or Birmingham? Tim will tell you the magical birds you can see there!), and frequent screenings of The Two Ronnies.

Is it Tim’s love of birds that caused him to title his gimungous pedagogical opus on cohomology, simplicial sets and crossed gadgetry, the Crossed Menagerie?

Tim gave the Friday morning talk at Barcelona. In the first hour, he began by saying that much of the material he had been wanting to cover had been touched on at various points in the conference - but that he wanted to highlight and point out various important constructions which are not as well-known as perhaps they should be. It was a very interesting talk, but I am not confident to write about it, because my simplicial skills are lagging behind somewhat (I seriously need a go at that menagerie). I’ll just mention the following topics which Tim felt important to stress: Kan complexes, the Moore complex of a simplical group, decollages, crossed complexes (and T-complexes), the “W bar” construction, and the Puppe sequence.

In his second, “presentation” talk, he presented the slides on Formal homotopy quantum field theories and 2-groups. Formal homotopy quantum field theories (HQFT’s) were invented by Turaev, and they are essentially TQFT’s in a background space $B$, up to homotopy. Rodrigues showed that an $n$-dimensional HQFT can be regarded as a monoidal functor

(1)$Z : nCob(B) \rightarrow Vect$

where $nCob(B)$ is the category whose objects are $(n-1)$ closed manifolds equipped with a map into $B$, and whose morphisms are cobordisms equipped with a map into $B$, considered up to homotopy (in $B$) fixing the boundary.

So… a HQFT is midway between an “abstract” TQFT (with no background space) and a Stolz-Teichner style “smooth” TQFT embedded in $B$. That makes them an important thing to understand.

Tim reviewed classification results for these gismos. For instance if $B$ is a $K(\pi, 1)$ for some group $\pi$, then Turaev showed that a 2d HQFT in $B$ is the same thing as a crossed $\pi$-algebra. My understanding is that a crossed $\pi$-algebra can be thought of as a Frobenius algebra object in $Rep \Lambda \pi$ - the category of representations of the loop groupoid of $\pi$. Is this correct?

Similarly Brightwell and Turner showed that if $B$ is a $K(A, 2)$, then a 2d HQFT over $B$ is the same thing as a Frobenius algebra equipped with an action of $A$. Again, my understanding is that this is the same thing as a Frobenius algebra object in $Rep A$. Is this correct?

Then Tim spoke about his work with Turaev, on extending these results to all 2-types. We know that a 2-type corresponds algebraically to a crossed module, so one begins by fixing a crossed module $\mathcal{C} = (C, P, \partial)$ and working from there. The main theorem is that formal 2d HQFT’s over a 2-type represented by $\mathcal{C}$ correspond to Frobenius algebras equipped with an action of the 2-group $\mathcal{C}$ by automorphisms.

The disclaimer “formal” is there to indicate that since they work directly with the 2-group $\mathcal{C}$ and a triangulation, they are not sure if they get out all HQFT’s in this way. Though I don’t understand the details, based on my experience with TQFT’s my rough intuition is that you should indeed generically get out all HQFT’s in this way - except possibly for the non-semisimple ones, which are non-generic, and can’t be achieved from a triangulation construction… is that right?

Anyhow something which was very interesting for me here is how an algebra can be acted on by a 2-group. I hadn’t thought of that before.

Posted at June 24, 2008 4:57 PM UTC

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### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

So… a HQFT is midway between an “abstract” TQFT (with no background space) and a Stolz-Teichner style “smooth” TQFT embedded in $B$.

It might be noteworthy that decategorifying the category of cobordisms with homotopy classes of maps into a certain space we get cobordism rings for cobordisms with the corresponding structure.

Oriented cobordisms for maps into $B SO$.

Spin cobordisms for maps into $B Spin$.

String cobordisms for maps into $B String$.

And so on. This does play an important role for instance in the stuff that Hopkins talked about, though he didn’t seem to be aware of Turaev’s concept of HQFT.

Posted by: Urs Schreiber on June 25, 2008 7:06 AM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

I am not sure I have sorted out how the following two concepts are meant to be related in the context of HQFT:

given a manifold $X$ (for instance a cobordism) and given an $n$-group $G$ with corresponding one-object $n$-groupoid $\mathbf{B}G$, there are two different types of $n$-functors to $\mathbf{B}G$ which one might consider.

1) For $Y \to X$ a surjective submersion and $Y^\bullet$ the corresponding Čech $n$-groupoid, $n$-functors $Y^\bullet \to \mathbf{B}G$ (or equivalently the corresponding simplicial maps, if you prefer)

2) For $T \subset X$ a triangulation of $X$, i.e. a $n$-cell complex modeling $X$, and $[T]$ the corresponding $n$-category, $n$-functors $[T] \to \mathbf{B}G \,.$

Homotopy classes of the maps in 1) give classes of $G$-$n$-bundles on $X$, hence homotopy classes of maps from $X$ to $B |G|$.

Whereas classes of the maps in 2) give classes of flat $G$-connections on $X$.

Posted by: Urs Schreiber on June 25, 2008 12:14 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

I’m not sure of the answer to Urs’ question. What I do know is that classically any triangulation gives rise to an open cover and hence to a simplicial sheaf (similar to the Cech n-groupoid). The best treatment I know of that sort of idea is probably Tibor Beke’s
Higher Cech Theory, (see http://faculty.uml.edu/tbeke/).

The question then is, sort of, whether there is a simplicial approximation theorem’ in this context. The fact that the manifold is locally contractible should be pencilled in somewhere so we can assume the open covers are Leray covers.

Posted by: Tim Porter on June 25, 2008 12:48 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

any triangulation gives rise to an open cover

By (maybe first passing to a dual triangulation and then) taking open neighbourhoods of all top-dimension cells and taking their disjoint union, right?

Doing that, an $n$-functor/simplicial map from the Čech groupoid of the cover thus obtained to some $\mathbf{B}G$ is not the same as a decoration of the original triangulation by $\mathbf{B}G$, it seems to me.

This is why I am not sure how I am supposed to think of this in the HQFT context.

But I need to have a closer look at some of the references…

Posted by: Urs Schreiber on June 25, 2008 3:04 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

The classical construction was to use what is called the star open cover of the triangulation. (I seem to remember it being in Spanier) The nerve of that open cover is then ISOMORPHIC (if I remeber it rightly) to the simplicial complex used for the original triangulation. There is a subsidiary result that given any open cover there is a triangulation finer’ than it. (This assumes that we start with a manifold.) By that is meant that the star open cover of the triangulation is finer than the given open cover. The definition of star open cover uses the vertex star of each vertex this being the union of the vertex and all interiors of all simplices of which it is a vertex. (Something like that..) The idea is discussed in sources which show that Cech cohomology and simplicial cohomology coincide. (Don’t trust my memory for the details!!!!!)

Posted by: Tim Porter on June 25, 2008 7:07 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

Thanks for the comment.

I believe I follow what you say so far. But are you claiming also that functors from the triangulation classify $G$-bundles?

Maybe it would help me to look at a simple example, the torus and the 2-sphere, maybe. Let’s pick the standard triangulation coming from starting with a square with sides identified and then cut in half diagonally to produce two triangles.

Let also $G$ be just an ordinary group.

Then $G$-colorings of the triangulation classify flat $G$-connections on that surface, but not $G$-bundles on that surface.

Maybe you can describe for me in terms of this example what the statement used in HQFT would be? I might still be misunderstanding something.

Posted by: Urs Schreiber on June 25, 2008 7:18 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

I will check up on things once I am back home. I should point out that the decomposition of the torus that you suggest is not a valid one as a triangulation, as it has only one vertex. The only simplicial complex with exactly one vertex is the zero simplex. (That should not disturb other things however as it is probably just a technicality.)

I am a bit hazy about the flatness issue hence need to do a bit of ferreting around. It may be worth looking in Turaev’s HQFT papers which are on the web, for instance math/9910010.

From my understanding of what he did, the key is a form of cellular approximation theory. Another important fact is that things are base point preserving, so the manifolds, taken as objects, have a chosen base point in each connected component, but the cobordisms with their characteristic maps are not thought of as being pointed, That is confusing but I think it just means that they are multipointed.

As I said I will check up. I have the feeling that the objects are manifolds with flat connection as you said.

One thing that I found confusing was that several authors think of the HQFTs as flat bundles on the base $B$, but we know a lot about $B$ if it is a $K(\pi,1)$ so why explore it like this. The degenerate case of TQFTs (i.e. with trivial base) tells you invariants of the manifolds and cobordisms, and that seems to me to be the more important aspect. I may of course be wrong!

Posted by: Tim Porter on June 26, 2008 8:04 AM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

I discussed this with Pietro Polesello who is also visiting Paris (and who gave a nice talk at Barcelona that no one has yet discussed, perhaps I will try to do so as Bruce has done more than me!)

After meeting accidently over lunch, we discussed further possible interpretations of the morphism of crossed modules that classified a formal HQFT. He had some interesting ideas but they need to be worked out more fully so I will say nothing more than that. However the one point that emerged (and had been staring me in the face) was that in the work I was reporting on the groups and crossed modules were discrete not Lie groups. This seems to be the key observation that I had missed. Does that resolve the difficulty?

Posted by: Tim Porter on June 26, 2008 6:45 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

discrete not Lie

Ah, I see. Right, I should have been aware of that.

Okay, so we expect that a 2-bundle with discrete 2-group has a unique connection, necessarily flat, and is uniquely characterized by that.

Saying so, I must admit that I realize that I need to think more closely about the details of the truth of this statement.

But David Roberts comes to the rescue. His work on 2-Covering Spaces should contain the answer to this question.

Posted by: Urs Schreiber on June 26, 2008 11:57 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

Okay, so we expect that a 2-bundle with discrete 2-group has a unique connection, necessarily flat, and is uniquely characterized by that.

Saying so, I must admit that I realize that I need to think more closely about the details of the truth of this statement.

But David Roberts comes to the rescue. His work on 2-Covering Spaces should contain the answer to this question.

As I have said before, one reason for my work on 2-covering spaces is to construct some 2-bundles globally (at the time I said explicitly, but that was probably the wrong word). For example, the universal 2-covering space - constructed using tricks from Urs’ and my paper and some other stuff already published by others. The question that came to me just yesterday was: “is there some sort of canonical connection on the universal 2-covering space?”

For this to even have a hope of being true, I have to find out the answer to this question: “is there some sort of canonical connection on the universal 1-covering space?” This should be well known if it is true. Presumably for discrete groups which have a finite dimensional smooth model for their classifying space* we can cook something up, but what about in general?

(* this includes things like torsion-free discrete subgroups of Lie groups - consider some sort of double coset space. My notes on this are not where I am, so take this with a little grain of salt)

Posted by: David Roberts on June 27, 2008 8:02 AM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

to construct some 2-bundles globally

By the way, just for the record: I suppose you know that Toby Bartels gives a general abstract method for (re-)constructing a (global in this sense) 2-bundle from its cocycle in the proof of his prop. 22. This general abstract prescription was recently spelled out by Christoph Wockel in the proof of his prop. I.20 (with comments on how it relates to Toby’s prescription in remark I.24).

I gather you want a less “by-gluing”-construction, but I thought I’d mention it anyway.

is there some sort of canonical connection on the universal 2-covering space?

For Lie 2-groups, you know that I have been claiming for a while now that there is – at least if you allow me to replace $B |G|$ by a rational approximation (but possibly you won’t allow me that :-): the $L_\infty$-algebraic notion of that universal connection is discussed in section 7.4 of $L_\infty$-connections and the way to integrate these to fully-fledged 2-bundles with connection I describe in On nonabelian differential cohomology.

But I haven’t really thought in detail about the case of discrete structure 2-groups. For those one would expect not only a canonical connection on the universal thing, but even a unique (and flat) connection on every single one of them.

Isn’t your discussion of 2-covering spaces particularly geared towards finite 2-groups?

Posted by: Urs Schreiber on June 27, 2008 12:55 PM | Permalink | Reply to this

### Re: Tim Porter on Formal Homotopy Quantum Field Theories and 2-groups

Was going to post this last night, but some piece of software somewhere was being silly.

I gather you want a less “by-gluing”-construction,

What I meant is that I want to construct some 2-bundles without using cocycle data at all.

I realise what a stupid question this is:

“is there some sort of canonical connection on the universal 1-covering space?”

Since the fibre is zero-dimensional so the tangent space at a point on the covering space has an obvious splitting - the trivial one!

It’s a bit trickier for 2-covering spaces since the fibres are not discrete groupoids, only weakly equivalent to them, and so the Lie-2-algebra the connection forms will take values in is not trivial, but equivalent to the trivial one.

And I’m not restricted to finite 2-groups - the easiest example I can think of has fibre $\mathbf{B}\mathbb{Z}$

Posted by: David Roberts on June 27, 2008 11:44 PM | Permalink | Reply to this

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