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February 27, 2009

Question on Geometric Function Theory

Posted by Urs Schreiber

I am further thinking about some issues which we discussed at the recent entry Ben-Zvi on geometric function theory (see nnLab: geometric function theory for some context).

From where I am coming the site over which we are looking at our generalized spaces in form of \infty-stacks is not an algebraic site and the (,1)(\infty,1)-stack QC of quasicoherent sheaves which Ben-Zvi-Francis-Nadler use so fruitfully as a model for nice geometric functions on generalized spaces is not manifestly available.

There is another construction naturally desiring to take its place, though, and I am wondering how the two perspectives would connect.

Suppose we have fixed some (,1)(\infty,1)-category C^\hat C whose objects we regard as generalized spaces (\infty-stacks), whose morphisms we regard as cocycles and whose 2-morphisms as coboundaries, etc, so that for XX, AA in C^\hat C we express the cohomology on XX with coefficients in AA as H(X,A):=π 0(C^(X;A))=Ho C(X,A)H(X,A) := \pi_0(\hat C(X;A)) = Ho_C(X,A).

A cocycle gC^(X,A)g \in \hat C(X,A) is interpreted as classifying an AA-principal bundle on XX, whose total “space” PXP \to X is the homotopy pullback of ptApt \to A along gg.

Suppose that C^\hat C sits inside an (,2)(\infty,2)-category C^\hat C' which hosts also the corresponding associated bundles (higher vector bundles) which may have non-invertible morphisms between them (which are 2-morphisms between cocycles=1-morphisms in C^\hat C'). A representation of AA is supposed to be some 1-morphism ρ:AVect\rho : A \to \infty Vect in C^\hat C' with Vect\infty Vect not in C^\hat C and the pullback along the composition of that with the cocycle gg gives the total space EE of the associated bundle.

Ordinary sections of EE are in Hom C^(X,Vec)(pt,ρg)Hom_{\hat C'(X,Vec)}(pt, \rho \circ g) and one can see that under homotopy pullback of the point every such section canonically gives rise to a span of total spaces of the form

Q Ω ptVect E \array{ && Q \\ & \swarrow && \searrow \\ \Omega_{pt} \infty Vect &&&& E }

where on the left we have the based loop object of Vect\infty Vect at the point at which we work.

Not all spans of this form arise from ordinary sections of EE this way: if we allow QQ here to be arbitrary such a span encodes general spaces UEU \to E over EE equipped with an ΩVect\Omega \infty Vect-valued cocycle on them.

To better see where we are, suppose we look at n=2n=2 where EE has as fibers 2-vector spaces and set Vect:=2Vect:=Ch(Vect)Mod\infty-Vect := 2Vect := Ch(Vect)-Mod in the above, equipped with the canonical point. Then ΩVect=Ch(Vect)\Omega \infty Vect = Ch(Vect) and an ΩVect\Omega \infty-Vect-cocycle is a complex of vector bundles.

So in this case generalized section spans as above arrange themselves naturally into a structure C(E)C(E) whose

- objects are pairs consisting of a space UEU \to E and a complex of vector bundles VUV \to U on UU;

- morphisms are pairs consisting of a commuting triangle U f U E \array{ U &&\stackrel{f}{\to}&& U' \\ & \searrow && \swarrow \\ && E }

together with a morphism Vf *vV \to f^* v' of the corresponding complexes of vector bundles. For other choices of Vect\infty Vect we get accordingly other structure than complexes of vector bundles on the spaces UU, UU'.

We may also forget for the time being the way we obtained the space EE here as the total space of some associated bundle and consider this construction C(X)C(X) for all objects (spaces) XX in C^\hat C.

Now I am coming to my question: it seems that this gadget C(X)C(X) wants to play the role of of the right notion of nice geometric functions on XX:

- whatever it is ((,1)(\infty,1)-category or the like), it is monoidal, thanks to the fact that it consists of maps to the (in general directed) loop space object ΩVect\Omega \infty Vect.

- it is naturally the thing acted on via pull(-tensor-)push by bi-branes, i.e. by those spans in C^\hat C'

Q E 1 E 2 \array{ && Q \\ & \swarrow && \searrow \\ E_1 &&&& E_2 }

which in the pull-push realization of QFT are supposed to act on them.

In fact, since C(X)C(X) should really be just the thing of span-morphism from ΩVect\Omega \infty Vect into XX, the action of further spans on this is much like the action of the category of spans on its under-category under ΩVect\Omega \infty Vect, if you see what I mean.

In any case, be that as it may: I am wondering how the “functions” C(X)C(X) for the special case spelled out above, i.e. for Vect:=Ch(Vect)Mod\infty Vect := Ch(Vect)-Mod would relate to the concept of “function” given by complexes of coherent sheaves.

From one point of view, C(X)C(X) is really the fibred category associated with the stack of complexes of vector bundles on the over-category C^/X\hat C/X. There is a canonical morphism from that to sheaves on XX. What is the image of that map when we are working over the algebraic site?

Posted at February 27, 2009 11:06 AM UTC

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Homotopy Limits in SSet Cat and QuasiCat

Here is another question, a straightforward one. The answer is certainly in one of the books on my desk, but maybe you can give me the answer before I find it myself:

Suppose CC is a category enriched in Kan complexes, KK is a small category regarded as a category enriched in discrete Kan complexes and j:KCj : K \to C is a KK-diagram in CC.

Then there is the homotopy limit over jj in its incarnation as a weighted SimpSetSimpSet-limit with weight W:KSSetW : K \to SSet, kN(K/k)k \mapsto N(K/k) (as described here).

On the other hand, we can regard CC as a quasi-category after passing to its simplicial nerve. Then there is Joyal’s definition of limit over jj as the terminal object of the over-quasi-category C /jC_{/j}.

I’d expect that these two notions of (homotopy) limit of jj coincide generally (if either exists). Is that right?

Posted by: Urs Schreiber on March 2, 2009 10:46 AM | Permalink | Reply to this

Re: Homotopy Limits in SSet Cat and QuasiCat

Yes; see section 4.2.4 in Lurie’s book.

Posted by: Mike Shulman on March 2, 2009 7:56 PM | Permalink | Reply to this

Re: Homotopy Limits in SSet Cat and QuasiCat

Yes; see section 4.2.4 in Lurie’s book.

Ah, right. Thanks.

Have now added this to nnLab: limit in quasi-categories.

Posted by: Urs Schreiber on March 2, 2009 9:05 PM | Permalink | Reply to this

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