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May 18, 2009

Journal Club – Geometric Infinity-Function Theory – Week 4

Posted by Urs Schreiber

In our journal club on [[geometric \infty-function theory]] this week Chris Brav talks about chapter 4 of Integral Transforms:

Tensor products and integral transforms.

This is about tensoring and pull-pushing (,1)(\infty,1)-categories of quasi-coherent sheaves on perfect stacks.

Luckily, Chris has added his nice discussion right into the wiki entry, so that we could already work a bit on things like further links, etc. together. Please see section 4 here.

Discussion on previous weeks can be found here:

week 1: Alex Hoffnung on Introduction

week 2: myself on Preliminaries

week 3: Bruce Bartlett Perfect stacks

Posted at May 18, 2009 7:08 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1968

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Re: Journal Club – Geometric Infinity-Function Theory – Week 4

Thanks Chris, elegantly written. Some quick questions for anyone out there, also posted on the nLab.

  • What’s a geometric stack? Is it just a ‘not-necessarily-perfect’ stack? Why call it geometric?
  • I have a feeling this question has been answered by David Ben-Zvi before. I’m suddenly a bit muddled up with the self-dual thing. Let’s decategorify to the setting of ordinary functions on compact complex manifolds. Let C(X)C(X) be the vector space of holomorphic functions on XX, and similarly for XX'. My understanding is that the analagous statement of geometric function theory is
    (1)Hom(C(X),C(X))C(X¯×X) Hom(C(X), C(X')) \cong C(\overline{X} \times X')

    where X¯\overline{X} refers to the complex manifold XX equipped with the ‘conjugate charts’… is this right? I’m not sure if XX¯X \cong \overline{X} for complex manifolds in general (maybe I’m just being dense here). So the self-dual step doesn’t go through in this setting. On the other hand, in the perfect stack situation everything is nicely self-dual. How come?

Posted by: Bruce Bartlett on May 18, 2009 10:22 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 4

Hi Bruce,

we talked about this by email: I am not sure where the X¯\bar X comes from. But let me make the following general comment, which should be obvious, but just so we are sure that we are all on the same page:

The statement that is being categorified here is the following very simple statement:

for XX and YY finite sets and for C(X)C(X) and C(Y)C(Y) the kk-vector spaces of kk-valued functions on XX and YY, the basic fact of matrix calculus says tjat there are natural isomorphisms of vector spaces

C(X)C(Y)C(X×Y)LinMaps(C(X),C(Y))C(X) *C(Y). C(X) \otimes C(Y) \simeq C(X \times Y) \simeq LinMaps(C(X),C(Y)) \simeq C(X)^* \otimes C(Y) \,.

It’s really that simple. The whole concept of “perfect derived stack” just says: consider a derived stack which happens to be well behaved enough such that the proof of this simple fact goes through literally for quasi-coherent sheaves!

When I read Integral transforms I do it in reverse order: the key theorem is prop. 4.6 on p. 28, which establish one piece of the above statement. Notice that the computation that proves this proposition looks like a trivial computation in linear algebra. It’s very simple. Then from there you can go back and say: okay, we want that this simple proof works, so now we declare XX and YY to be objects nice enough so that it does work. So XX and YY are to be perfect stacks, which pretty much says nothing but that: on them this simple linear algebra style reasoning makes sense literally.

Posted by: Urs Schreiber on May 19, 2009 7:23 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 4

What’s a geometric stack?

That’s a stack XX which has a cover YXY \to X where YY is an “ordinary space”, or ´rather, where YY is a representable stack.

So if the underlying site is that of topological spaces, YY is required to be a topological space and, geometric stacks are then called [[topological stack]].

If the underlying site is one of algebras opalgebras^{op}, then YY is required to be a dualized algebra (an affine algebraic space), and a geometric stack is called an [[algebraic stack]].

These nnLab entries with lots of details have kindly been created by Zoran Škoda yesterday.

(But yesterday some software bug prevented me from posting this here, for some reason.)

Posted by: Urs Schreiber on May 20, 2009 8:11 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 4

Is it already Tuesday?

I was busybusybusy. Chris Brav is travelling. He managed to fill in a bit more on section 5 “Applications” before he had to leave. Have a look, it’s good stuff.

But I guess I should be waiting with the next week’s installment of the journal club maybe until next week.

The further plan is that, if I remember correctly, Alex Hoffnung then collects the total wisdom absorbed at Northwestern on TQFTs and gives us a lecture on section 6.

Then, after that, we take a short break, look through the material, start fervently discussing the material, and then continue with “Character Field Theory”.

I just come from a talk by Catharina Stroppel, which ended with the question: “Where do we go from here?” and the answer: “We follow Ben-Zvi/Francis/Nadler, albeit just abstract nonsense.” :-)

So that’s what we’ll do.

Myself, next week I’ll be on vacation, then the week after that in Oberwolfach, then Bruce Bartlett and Igor Baković visit Hamburg, then June is almost over and mid of July is approaching, from when on I’ll have more leisure.

Posted by: Urs Schreiber on May 26, 2009 4:43 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 4

Hi

I am a little behind on section 6. I just started doing it today honestly, but I am about half way done. So I will post soon.

Posted by: Alex Hoffnung on June 2, 2009 9:54 AM | Permalink | Reply to this

(infinity,1)-operads

I seem to remember that somewhere in our journal club discussion – but I just can’t find it right now – David Ben-Zvi had mentioned that motivated by consideratins as in John Francis’ thesis Jacob Lurie was going to rewrite his article Commutative algebra in terms of higher operads.

Apparently this has happened by now. See the updated and commented list of references here.

I briefly wrote a stub entry: (,1)(\infty,1)-operad.

Posted by: Urs Schreiber on September 16, 2009 10:17 PM | Permalink | Reply to this

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