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April 27, 2009

Journal Club – Geometric Infinity-Function Theory – Week 1

Posted by Urs Schreiber

In our Journal Club on geometric \infty-function theory this first official week starts with Alex Hoffnung talking about section 1 of “Integral Transforms”.

This is to get us going and hopefully also reduce the intimidation level. If it looks interesting, have a look at our schedule. We are still looking for volunteers who would like to have a look at section 4,5, and 6 of “Integral Transforms” and write some kind of report for us all, to start further discussion.

Journal club post by Alex Hoffnung


Hi-

I have not been asked to write a “book report” since 3rd or 4th grade. Somehow this was more difficult than I remember. Anyway, I hope the following serves as a useful jump off for our journal club.

I am going to attempt to report on the introductory sections of the Ben-Zvi, Francis, Nadler (BZFN) paper. So how does one describe an introduction which already gives a detailed section by section description of the whole of the paper? Let’s find out. (Here is a hint: restate or copy shamelessly much of what the authors have already kindly told us.)

Let me start by saying what BZFN are attempting to do. The main applications stated are in the simpler form the calculation of Drinfeld centers of monoidal categories of sheaves and construction of topological field theories.

The reason to first state the applications which seem to be way off in our future is just to get down the jargon and then put it to the side for a while, hopefully lessening our burden while we trudge through the preliminaries.

So what is a Drinfeld center (of a monoidal category (of sheaves)) and roughly how does one calculate it? I guess (from reading the abstract) that the Drinfeld center should be an \infty-subcategory of the category of quasicoherent sheaves and is the same as the Hochschild cohomology category. Of course, this still leaves a lot of explaining to do. Hopefully, someone who wants to report on Section 5 can jump in here and explain this. If not, I will come back later and try to understand/explain these things.

Second, what are topological field theories (TFT’s) and roughly how does one construct such a thing? A TFT is a symmetric monoidal functor from a certain cobordism \infty-category to a symmetric monoidal \infty-category. In Section 6, we will see an explicit construction of 2d TFT where the circle is sent to the \infty-category of quasicoherent sheaves on the loop space of a perfect stack. Similarly, I hope this can be a jumping off point for someone to talk about Section 6.

Having vaguely introduced the applications, what are the main results that this paper will present as our tools? We want to identify the category of sheaves on a fiber product with two algebraic constructions:

i) the tensor product of the categories of sheaves on the factors,

and

ii) the category of linear functors between the category of sheaves on the factors.

Here the authors tell us that (ii) allows us to realize functors as integral transforms. So far I have been laying out an outline for this paper where we consider these factors mentioned above to be schemes or stacks (except maybe briefly in the applications). Lets quickly finish the story for this case, make note of the fact that we want to take everything above and make it a bit fancier by considering derived stacks, then use the comment above on (ii) to consider an even simpler example which DBZ was kind enough to explain in a previous post. Hopefully, this will help us find some intuition for why and how we want to understand (i) and (ii) before getting scared off by things like derived stacks.

If we let X be a scheme or a stack, then we can obtain a stable \infty-category QC(X), whose homotopy category is the unbounded quasicoherent derived category D qcD_{qc}(X). QC(X) becomes a symmetric monoidal stable \infty-category. So we want to calculate the inftyinfty-categories built out of QC(X) from tensor products, linear functors and other algebraic constructions in terms of geometry on X.

So that last paragraph (taken almost word for word from the paper) is about where my boundaries begin to be pushed. Now I think the best thing to do is take a step back and remember that David was kind enough to write us a short note on geometric function theory a while back. In the interest of keeping this self-contained and accessible to as many people as possible (including me) we should recap the lessons of this note.

The toy model: function theory on finite sets. This is a great introduction to some of the main characters in our story, in particular, spans or correspondences, the push-pull, integral transforms, functions on (fiber) products and maps between different structures of functions on algebraic objects (in this case sets). Eventually functions will mean quasicoherent sheaves and sets should mean something like scheme, stack, groupoid, or derived stack.

We start by considering a span of finite sets

(1)XZY X\leftarrow Z \rightarrow Y

and, for instance, the vector spaces of complex functions on each of these sets. It is very natural to pull back a function on X to a function on Z by the function from Z to X. Fortunately, we can also push-forward a function on Z to a function on Y by summing over fibers, allowing us to obtain linear maps from [X]\mathbb{C}[X] to [Y]\mathbb{C}[Y]. This turns out to be just a special case of a richer phenomenon. What I have just described doesn’t fully exploit the span. We can choose a function on Z and when summing over fibers of elements of Y convolve by this function, thus obtaining a new linear map from [X]\mathbb{C}[X] to [Y]\mathbb{C}[Y]. My first description implicitly involved convolution with the constant function 11 on Z. So, in general, functions on Z are integral transforms taking functions on X to functions on Y for any span of the sort given above.

Now, Ben-Zvi goes further to explain the how products play into this game. (I should mention that this story of the toy model was first told to me by Tony Licata) This will give us a nice base to think about the more difficult theorems we are attempting to understand in this journal club. But first, I would like to at least make a small excursion to mention the relation to groupoidification (since I have a vested interest in the subject and it has been discussed here and here extensively). This toy model which we intend to extend to geometric \infty-function theory is also a toy example for groupoidification. Spans of sets are just really boring examples of spans of groupoids and using the concept of groupoid cardinality described at the links above and zeroth (co)homology of groupoids, we can employ the same pull-push operation and convolution to obtain linear maps. Also, one could continue reading Ben-Zvi’s note to consider functions on finite orbifolds and leading up to his own synopsis of the paper we are currently reading.

Lets understand Ben-Zvi’s comments on products and then get back to the paper at hand. Given finite sets X and Y, there is of course a nice span

(2)XX×YY X \leftarrow X \times Y \rightarrow Y

and using the construction above we get

(3)[X×Y]Hom ([X],[Y])\mathbb{C}[X\times Y] \cong Hom_{\mathbb{C}}(\mathbb{C}[X],\mathbb{C}[Y])

between integral kernels and linear transformations.

When we have a cospan

(4)XYX X \rightarrow Y \leftarrow X'

we get a relative version of this construction

(5)[X× YX]Hom [Y]([X],[X]) \mathbb{C}[X\times_{Y} X'] \cong Hom_{\mathbb{C}[Y]}(\mathbb{C}[X],\mathbb{C}[X'])

.

The theorems we intend to try to understand use an exact analogue of this correpondence.

So now we should take the plunge into derived stacks and try to repeat the story just told.

1) stacks and higher stacks arise from quotients (or more general colimits) on schemes

2) derived stacks arise from fiber products (or more general limits) on schemes and stacks

I would like to eventually include some links to definitions and examples here. Any help with that would be great.

Now we begin repeating what we have said above in this derived setting. To a derived stack X, we assign a stable \infty-category QC(X) in a manner which extends the definition for an ordinary stack or scheme.

In particular, let X = Spec R, an affine derived scheme, then QC(X) is the \infty-category of R-modules Mod RMod_R whose homotopy category is the usual unbounded derived category of R-modules. As before, the tensor product provides QC(X) with the structure of a symmetric monoidal stable \infty-category.

Now we should state the generalized versions of the previous stated applications and main results. The main applications being the calculation of the Drinfeld centers (and higher E n\E_n-centers) of \infty-categories of sheaves and functors. BZFN give as an example, the identification of the Drinfeld center of the quasicoherent affine Hecke category with sheaves on the moduli of local systems on a torus. And as a second application, explaining how the results obtained fit into the framework of 3-dimensional topological field theory (of Rozansky-Witten type). In particular, BZFN verify categorified analogues of the Deligne and Kontsevich conjectures on the E n\E_n-structure of Hochschild cohomology.

We are clearly back to the land of scary. So I think the best thing to do is to follow BZFN and give a brief overview of each section, which can serve as an outline to be filled in over the course of this journal club.

Why perfect stacks? Urs has begun to tackle this question here . We will be mainly interested in studying the \infty-category QC(X)QC(X), for X a derived stack, but this is in general too unwieldy algebraically. The problem being that it may contain objects that cannot be constructed in terms of concrete, locally finite objects. The “perfect” solution is to consider a smaller \infty-subcategory QC(X) QC(X)^\cdot of “generators” which are “finite” in some sense. This leads us to the idea of an ind-category. This is a nice idea, where one considers a small category of “managable” objects which has diagrams whose inductive limits are morally the larger objects which may not be quite so managable. Of course, these objects cannot be limits of any sort since they do not live in this smaller “managable” category, so one considers the diagrams themselves as placeholders for these objects. There is another apporach which involves the vanishing of right orthogonals, and I do not really understand this. Maybe someone can step in here and give me a hand.

In Section 3.1, BZFN discuss appropriate notions of “finiteness”

-perfect objects (geometry)

-dualizable objects (monoidal structure)

-compact objects (categorical structure)

I would like to discuss how these relate to the corresponding notions in parenthesis, but I am not quite sure how this goes yet.

A quasicoherent sheaf MQC(X)M\in QC(X) is a PERFECT COMPLEX if it locally restricts to a perfect module (a finite complex of free modules). Equivalently, M is DUALIZABLE with respect to the monoidal structure on QC(X). We (or maybe I) should probably create nlab pages for the capitalized words.

We are now in a position to define a PERFECT derived stack X. The two conditions are that it has an affine diagonal and that QC(X) is given as the inductive limit of the full \infty-subcategory of perfect complexes. Then we can, of course define PERFECT morphisms.

On a perfect stack X, compact objects of QC(X) are the same as perfect compexes (which are the same as dualizable objects).

I think there should be some discussion on the importance of compact objects. Hopefully, I can come back and include that here soon.

So we can reformulate the definition of a perfect derived stack as having an affine diagonal, QC(X) being compactly generated (i.e. no right orthogonal to the compact objects) and that compact and dualizable objects coincide.

BZFN remark that compactly generated categories can be expressed as categories of modules.

Moving on to tensors and functors (Section 4) we can see analogues of the constructions in the toy example. For X, X’ schemes over a ring k, the dg category of k-linear continuous (colimit preserving) functors is equivalent to the dg category of integral kernels:

(6)Fun k(QC(X),QC(X))QC(X× kX) Fun_k(QC(X),QC(X')) \cong QC(X\times_k X')

(a theorem of Toen). A similar result holds for dg categories of perfect (equivalently, bounded coherent) complexes on smooth projective varieties.

The main technical results are as follows:

- the tensors and functors of \infty-categories of quasicoherent sheaves are calculated in the symmetic monoidal \infty-category Pr LPr^L of presentable \infty-categories with morphisms left adjoints (Section 2)

- the tensors and functors of \infty-categories of erfect complexes are calculated in the symmetric monoidal \infty-category Idem of k-linear idempotent complete stalk small \infty-categories (Section 4.1)

Finally we want to state the theorems:

For XYXX \rightarrow Y \leftarrow X' maps of perfect stacks, there is a canonical equivalence

(7)QC(X× YX)QC(X) QC(Y)QC(X) QC(X\times_Y X') \cong QC(X)\otimes_{QC(Y)} QC(X')

between the categories of sheaves on the derived fiber product and the tensor product of the \infty-categories of sheaves on the factors.

There is also a canonical equivalence

(8)Perf(X×X)Perf(X)Perf(X) Perf(X\times X') \cong Perf(X)\otimes Perf(X')

for inft\inft-categories of perfect complexes.

For XYX\rightarrow Y a perfect morphism to a derived stack with affine diagonal and XYX' \rightarrow Y arbitrary, there is a canonical equivalence

(9)QC(X× YX)Fun QC(Y)(QC(X),QC(X)) QC(X\times_Y X') \cong Fun_{QC(Y)}(QC(X),QC(X'))

between the \infty-category of sheaves on the derived fiber product and the \infty-category of colimit-preserving QC(Y)-linear functors.

When X is a smooth and proper perfect stack, there is also a canonical equivalenc

(10)Perf(X×X)Fun(Perf(X),Perf(X)) Perf(X\times X') \cong Fun(Perf(X),Perf(X'))

for \infty-categories of perfect complexes.

The rest of this section gives an overview of the applications of these results. We will get to this soon.

I have attempted to recap what I read in the introduction here. My main goal was to provide a template for our journal club. I hope that we can transport parts of this over to the nLab and fill in the blanks will more detailed exposition as we go. I had some technical trouble such as using subscripts. Maybe someone can tell me what I am doing wrong.

Posted at April 27, 2009 4:48 PM UTC

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Re: Journal Club – Geometric Infinity-Function Theory – Week 1

This is Bruce checking in for the Monday Seminar. Thanks Alex, that’s a great template for a first post, and I like the undercurrent of humour in your style!

The good news about this Introduction section is that (a) as you say, David Ben-Zvi has kindly sketched out the basic idea in the geometric function theory notes, and (b) there is a “Preliminaries” section which describes in a bit more detail (but not totally scary detail) the stuff which was described in this Introduction section. Urs will be covering this section next week Monday.

While I look up stuff like “ind-category” and “right-orthogonal” on the nLab, let me say what I got out this time from reading the Introduction section again: the last section on Rozansky-Witten theory was great, and solved some conceptual puzzles I’ve always had — at least, one day when I truly understand what things like “Spec Sym Ω X[1]\Omega_X[1]” are.

But simply put, what always confused me with Rozansky-Witten theory was the following. Rozansky-Witten theory is meant to be a 3d extended TQFT based on a nice complex manifold XX. So it assigns a category to the circle. From abstract 2-categorical nonsense, we believe that the category assigned to the circle in an extended TQFT should be the “center” or “dimension” or “looping” of the 2-category assigned to the point:

(1)Z(S 1)dimZ(pt). Z(S^1) \cong dim Z(pt).

Why do we believe that? Well, just because the circle looks like a little “loop”.

In Rozansky-Witten theory, the category assigned to the circle is the derived category of coherent sheaves on XX:

(2)Z(S 1)=D(X). Z(S^1) = D(X).

That’s a bit weird though, because we know that Z(S 1)Z(S^1) is supposed to be some kind of looping of Z(pt)Z(pt), and we don’t see any “loops” in D(X)D(X). What’s going on?

In the paper we are learning about, David Ben-Zvi, John Francis and David Nadler show that there is another theory which I guess is what happens if you plug in the basic ideas of Rozansky-Witten theory into their big machinery and turn the crank. What pops out is not exactly Rozansky-Witten theory. But the thing it assigns to the circle is very closely related to what ordinary Rozansky-Witten theory assigns to the circle. In fact, we have:

(3)Z BenZvi,Francis,Nadler(S 1)=Z RozanskyWitten(S 1) Z_{Ben-Zvi, Francis, Nadler} (S^1) = Z_{Rozansky-Witten} (S^1)

up to “completion of the latter along the zero section”. The great thing about this new theory though is that it has all the “loops” correctly filled in, and in fact the authors will soon prove that Z(S 1)Z(S^1) represents the “looping” of the 2-category assigned to the point in an upcoming paper. The bad thing about it is that apparantly it only works at the level of 0, 1- and 2-manifolds; it can’t quite be defined for 3-manifolds for reasons I don’t understand.

Posted by: Bruce Bartlett on April 27, 2009 5:01 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 1

Thanks, Alex!

I’ll try to reply in more detail later. For the moment just this small comment:

Now I think the best thing to do is take a step back and remember that David was kind enough to write us a short note on geometric function theory a while back.

Yes! So one main ingredient here is that we we want to be looking at the \infty-version of finite-dimensional linear algebra under the following disctionary:

- finite sets SS are replaced by generalized spaces aka derived \infty-stacks XX;

- vector spaces [S,][S,\mathbb{R}] of functions S S \to \mathbb{R} on finite sets are replaced with (,1)(\infty,1)-categories [X,QC][X, QC] of some kind of \infty-vector bundles – called QCQC here! –

- we prove that as linear maps between finite dimensional vector spaces [S,][S,][S,\mathbb{R}] \to [S',\mathbb{R}] are given by matrices, suitably well behaved \infty-functors [X,QX][X,QC][X,QX] \to [X', QC] are given by “\infty-matrices” called integral transforms.

- and then we check how the familiar notions of trace and center generalize from finite dimensional endomorphisms to such integral transforms and find that they yield lots of nice relations to Hochschild cohomology.

Have to run now…

Posted by: Urs Schreiber on April 27, 2009 5:05 PM | Permalink | Reply to this
Read the post The Earth - For Physicists
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Tracked: April 28, 2009 1:20 AM

Re: Journal Club – Geometric Infinity-Function Theory – Week 1

Now finally have a bit of time to look into this. I’ll read Alex’s report and comment where comments come to mind.

So here goes:

I guess (from reading the abstract) that the Drinfeld center

We need to start an nnLab-entry [[Drinfeld center]].

should be an \infty-subcategory of the category of quasicoherent sheaves

We might maybe remark that the construction is more general:

For every(,1)(\infty,1)-monoid” defined to be an algebra object AA in a [[closed monoidal symmetric (,1)(\infty,1)-category?]] there is a notion of [[derived center?]]

Z(A)=End AA *(A) Z(A) = End_{A \otimes A^*}(A)

and this generalizes/expands the notion of [[Drinfeld center?]] of a monoidal category.

Eventually (like from page 34 on) we are interested in setting

A=Vect(X), A = \infty Vect(X) \,,

the (,1)(\infty,1)- (or is it (,0)(\infty,0)-?)category of some flavor of \infty-vector bundle-like things on a generalized space XX.

The center does (see page 33 of “Integral Transform” and page 19 of “Character Theory”) come with a canonical morphism

Z(A)A. Z(A) \to A \,.

Is that an inclusion in a suitable sense? One would hope it is, but I don’t know at the moment. Do you?

Hopefully, someone who wants to report on Section 5 can jump in here and explain this.

Yes, that would be great. Now that we already have a fourth volunteer doing section 4!

By the way, I had made some feeble attempts to prepare the ground for this section 5 a bit by creating

span trace

and

cospan cotrace

to be read eventually in the context of

loop space object

and

homotopy limit.

So we want to calculate the infty-categories built out of QC(X) from tensor products, linear functors and other algebraic constructions in terms of geometry on X.

So that last paragraph (taken almost word for word from the paper) is about where my boundaries begin to be pushed.

Right, this is one thing we want to sort out in detail eventually.

Just for the record, notice that an attempt at a clean crisp statement about

built out of QC(X)QC(X) from tensor products, linear functors and other algebraic constructions

is at nnLab: geometric function theory, when you scroll down to the section “general \infty-categorical / homotopical setup”.

I don’t think it will be too hard to get this under reasonable detailed control eventually.

It’s to a large extent the familiar symbol manipulation with the main and crucial new thing that keeps coming up again and again the fact (in one incarnation or other) that the homotopy pullback of a point in XX along itself is not the point, as it is in the 1-dimensional theory, but is loops in that space

ΩX * * X. \array{ \Omega X &\to& * \\ \downarrow && \downarrow \\ * &\to& X } \,.

Or equivalently: the fact that we are allowed to replace whenever we are to compute a pullback all points by big puffed up contractible spaces, which we may think of as total spaces of generalized universal bundles.

We start by considering a span of finite sets

Cool. By the way, why does no one from the groupoidification crew feel like adding some introduction and statements about it at groupoidification? Maybe just copy-and-pasted from existing introductions hidden somewhere on the blog?

That would have saved you from typing the next few paragraphs.

Well, or the other way round: some paragraphs like this should eventually be copied and transmuted into that nnLab entry. I think.

1) stacks and higher stacks arise from quotients (or more general colimits) on schemes

2) derived stacks arise from fiber products (or more general limits) on schemes and stacks

I would like to eventually include some links to definitions and examples here. Any help with that would be great.

Some definitions exist on the Lab, more examples and more everything surely would be nice to add. There is

\infty-stack (though that needs rewriting, but follow some of the links)

then

derived \infty-stack

which I just notice has an almost-duplicate at derived stack.

A bird’s eye starting point is (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves

(where (,1)(\infty,1)-sheaf = \infty-stack!).

Well, I also tried to write an introduction to that stuff at heuristic introduction to sheaves, cohomology and higher stacks.

(Most important when looking at these entries: there is lots of chance to get the feeling that this is suboptimal and missing something. If anyone begins to feel this way: edit the entries to improve them!)

-[[perfect objects?]] (geometry)

-[[dualizable objects?]] (monoidal structure)

-nnLab: compact objects (categorical structure)

I would like to discuss how these relate to the corresponding notions in parenthesis, but I am not quite sure how this goes yet.

Oh, this is probably not meant to be anything deep:

- dualization is something you do in a monoidal category. So there goes the monoidal.

- Compact objects are defined in terms of homs out of them respecting certain colimits, so that’s category-theoretic.

- And those perfect stacks are some kind of \infty-vector bundles that locally look like finite complexes of ordinary vector bundles. So that’s a geometric, i.e. bundle-theoretic description.

By the way, David Ben-Zvi made some helpful comments on perfect stacks here in the comment section of his original guest post.

derived stack as having an [[affine diagonal?]]

I’ll skip over the nice summary of theorems and results… about to suggest that

we can transport parts of this over to the nLab and fill in the blanks will more detailed exposition as we go.

Yes!

That list of theorems and results of “Integral Transforms”, for instance, should be, to get started, copy-and-pasted straight to the Journal Club entry. That’s where it belongs.

Similarly other parts of your report could be copy-and-pasted to the relevant entries (and these entries created if necessary) for a start. Polishing will come by itself then, as people work on entries.

Okay, I am hoping I am not getting on your all nerves with this insistence of moving material to the nnLab. But it’s the better for all of us.

The Journal Club entry has already placeholder headlines for all the relevant sections of the BZNF articles. Eventually each of them should be filled with a one or two or a handful of paragraphs in just this style of your report here.

Posted by: Urs Schreiber on April 28, 2009 8:54 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 1

Thanks for the great book report Alex!

One quick comment: the Drinfeld center is not a “sub” in any reasonable sense (except in the loose sense that it’s a categorical limit). This is easy to see already for the category of representations of a finite group, where the Drinfeld center (or modules for the Drinfeld double) is the category of G-equivariant vector bundles on the group G (I think they also go by the name Yetter-Drinfeld modules?) Likewise the center of a ring in the derived world (ie Hochschild cohomology) is not really a subring - even for a commutative ring, its derived center doesn’t map in injectively (for a smooth commutative ring the Hochschild-Kostant-Rosenberg theorem tells us that Hochschild cohomology is the exterior algebra on derivations of the ring). I think we have to abandon the notions of sub and quotient in the homotopical world and stick to notions like (homotopy) limit and colimit.

I hope to add some comments later, but am busy getting married at the moment..

Posted by: David Ben-Zvi on April 29, 2009 7:14 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory – Week 1

Hi David

I am about to hop on a plane so will try to respond with actual content and expand my post later. I heard you were getting married so I figured you would be busy. Thanks for taking the time to respond and congratulations!

Posted by: Alex Hoffnung on April 30, 2009 4:24 AM | Permalink | Reply to this

Week 2, section 2

According to our schedule, today is my turn to talk in our Journal Club, this time about section 2 “Preliminaries” of “Integral Transforms”.

But I will be late one day and will instead post my part tomorrow.

The reason is that I don’t yet have my piece on section 2.5 worked out. And also the material I have needs a bit more polishing before I want to oficially post it. And today I am busy with something else.

But most of what I will post tomorrow is already visisble under “section 2” at our nnLab web page. It’s main purpose is to be a commented list of links to nnLab entries with more details, anyway.

So everybody is kindly invited to look at the nnLab entry to find four fifths of the material for the second week of our Journal Club. Corrections, questions and improvements are welcome, especially if typed right away into the nnLab entry.

So, see you tomorrow. Sorry for the delay.

Posted by: Urs Schreiber on May 4, 2009 8:40 AM | Permalink | Reply to this

Re: Week 2, section 2

Ok, thanks Urs!

Posted by: Bruce Bartlett on May 4, 2009 10:20 AM | Permalink | Reply to this
Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 2
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Excerpt: Preliminaries for the discussion of geometric infinity-function theory: higher categories, higher sheaves, higher algebra, higher traces and what it all means.
Tracked: May 5, 2009 3:52 PM
Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 3
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Excerpt: This week in our Journal Club on [[geometric ∞-function theory]] Bruce Bartlett talks about section 3 of "Integral Transforms": perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for...
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Excerpt: Chris Brav reviews technical details about tensor products and integral transforms of quasi-coherent sheaves on perfect stacks.
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