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June 3, 2009

Journal Club — Geometric Infinity-Function Theory — Week 6

Posted by John Baez

guest post by Alex Hoffnung


This section is broken into two short parts. I will try to say a little about the first here and then head over to the nnLab to say more soon.

Topological field theory from perfect stacks

Early on we learn about 22-dimensional TQFT’s as functors from 2Cob2Cob to HilbHilb or something of that sort. Here 2Cob2Cob is the category whose objects are 11-dimensional manifolds and whose morphisms are 22-dimensional manifolds whose boundary we think of as having an “incoming” 11-dimensional manifold as source and similarly an “outgoing” 11-dimensional manifold as target. Of course, HilbHilb could have been replaced with a variety of different categories.

Eventually we grow up and have to learn new things. So today we are going to change our notion of 2Cob2Cob: let 2Cob2Cob denote the \infty-category whose objects are compact oriented 11-manifolds and whose morphisms 2Cob(C 1,C 2)2Cob(C_1,C_2) consist of the classifying spaces of oriented topological surfaces with fixed incoming and outgoing components C 1C_1, C 2C_2. This is given symmetric monoidal structure by disjoint union of 11-manifolds.

Then of course we need to replace HilbHilb by a symmetric monoidal \infty-category 𝒞\mathcal{C}.

Now if we know what these symmetric monoidal \infty-categories are then we are ready to define the notion of topological field theory. Since all the hard work was pushed into understanding 2Cob2Cob, then the definition is obvious:


A 22-dimensional topological field theory is a symmetric monoidal functor F:2Cob𝒞F: 2Cob \rightarrow \mathcal{C}.

So that was the background that we need. Maybe in a few years we will say “Early on we learn about 22-dimensional TFT’s as symmetric monoidal functors from 2Cob2Cob to 𝒞\mathcal{C}” and everyone will know that we mean \infty-categories.

Next we define the values of our TFT’s for the present example. We first fix a derived ring kk.


Let Cat exCat_\infty^{ex} denote the (,2)(\infty,2)-category of stable, cocomplete kk-linear \infty-categories, with 11-morphisms consisting of continuous exact functors.

So when I hit ‘(,2)(\infty,2)-category’ something funny happens and the rest of the sentence comes out as “…of blah-blah-blah-categories, with 11-morphisms consisting of continuous exact functors.” This shouldn’t really happen since stable, cocomplete and kk-linear are friendly enough concepts, but it definitely tells me I should slow down and learn to love (,2)(\infty,2)-categories.

I am going to again put off explanations for a later date at the nnLab.

Thinking about the symmetric monoidal structure for a moment gets us to the first of our two results which are approximately of the form “You give me a perfect stack and I will give you a TFT”.

The symmetric monoidal structure comes from the tensor product of presentable stable categories with unit the \infty-category of kk-modules. Why do we care? From here we can see what the TFT should assign to closed surfaces. In our example the answer will just be a kk-module.

How does it work? Luckily, this is ‘monoidally easy’ so I can hopefully understand it. We note that the unit of 2Cob2Cob is the empty 11-manifold, i.e., the boundary of a closed surface. So a TFT sends closed surfaces to endomorphisms of the unit of the target \infty-category 𝒞\mathcal{C}. For our choice 𝒞=Cat ex\mathcal{C} = Cat_\infty^{ex}, we obtain our previous answer of a kk-module as the image of a closed surface, since the unit is Mod kMod_k.

Now someone should give me a perfect stack called XX and we can state our theorem:


For a perfect stack XX, there is a 22-dimensional TFT 𝒵 X:2CobCat ex\mathcal{Z}_X: 2Cob \rightarrow Cat_\infty^{ex} with 𝒵 X(S 1)=QC(X)\mathcal{Z}_X(S^1) = QC(\mathcal{L}X) and 𝒵 X(Σ)=Γ(X Σ,𝒪 X Σ)\mathcal{Z}_X(\Sigma) = \Gamma(X^\Sigma,\mathcal{O}_{X^\Sigma}), for closed surfaces Σ\Sigma.

Then there is a proof. I am not really going to read that right now, even though it seems like that is where the goodies are. For now let’s note some important points:

  • Since all of the stacks involved are perfect, pullback and pushforward of quasicoherent sheaves along this correspondence defines a colimit-preserving functor.

Now I have not told you what the correspondence is, but we can still see why it is so nice to be perfect. A big theme has been the quest for function theories with nice pullback and pushforward conditions. In particular, that integral transforms should be the same as cocontinuous functors.

Second we note again the symmetric monoidal structure on 𝒵 X\mathcal{Z}_X. In particular, our other favorite fact comes into play:

  • QC(X 1) QC(Y)QC(X 2)QC(X 1× YX 2)QC(X_1)\otimes_{QC(Y)}QC(X_2) \stackrel{\sim}\rightarrow QC(X_1\times_Y X_2),

where there are maps of perfect stacks p 1:X 1Yp_1: X_1 \rightarrow Y and p 2:X 2Yp_2: X_2 \rightarrow Y.

It seems that we are all somewhat starving for examples of perfect/derived stacks. Maybe we are just looking too hard! BZFN give a nice example here of a TFT coming from the classifying space BGBG.


Let X=BGX = BG (in characteristic zero). Then the corresponding TFT assigns to the circle the \infty-category of sheaves on the adjoint quotient G/G=Loc G(S 1)G/G = Loc_G(S^1). That is, GG-local systems on the circle. To a surface Σ\Sigma, the TFT assigns the cohomology of the structure sheaf of the moduli stack BG Σ=Loc G(Σ)BG^\Sigma = Loc_G(\Sigma).

Again there is more to be said here but it will have to wait to get to the lab.

They note that the 22-dimensional topological field theory defined above is a categorified analogue of the 22-dimensional TFT’s defined by string topology on a compact oriented manifold, or the topological BB-model defined by a Calabi–Yau variety.

I am not at all qualified to comment on this statement without doing a little digging, and since I am short on time I will just wait and see if someone else chimes in here.

BZFN note that the their TFT operations are not sensitive to smooth structure. They then go on to show how the proof of the existence of a TFT given a perfect stack can be extended to give proof of the existence of a TFT on spaces as well. This is the second proposition/theorem.

I will quit here and send this over to the nnLab for further work. Also, I will begin to say something there about the Deligne–Kontsevich conjectures for derived centers.

Posted at June 3, 2009 5:57 AM UTC

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Re: Journal Club — Geometric Infinity-Function Theory — Week 6

Nice job Alex. I still have a lot of catching up to do before I can join the discussion but I haven’t forgotten about it.

Posted by: Christoph Sachse on June 3, 2009 10:29 PM | Permalink | Reply to this

Re: Journal Club — Geometric Infinity-Function Theory — Week 6

Thanks, Alex, for this!

And thanks to John for posting it here. I have been on vacation and offline last week without telling you all. But now I am back. Will try to catch up with what happened here.

On the other hand, tonight I’ll take a night train to Oberwolfach for next week’s workshop “Strings, Fields and Topology”. While on-topic for our journal club, I’ll have to see how much spare time I find myself left with…

Posted by: Urs Schreiber on June 7, 2009 4:05 PM | Permalink | Reply to this

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