Maltsinotis: Grothendieck and homotopical algebra

More notes from the Grothendieck conference: Maltsiniotis spoke on Grothendieck and homotopical algebra – here is my scan. In my opinion this talk was one of the best, given that the subject is often regarded as quite inaccessible, and still he managed to make it very enjoyable. The talk covered basic notions and ideas from Grothendieck’s work in the 80s, when he was thinking about stacks, higher categories, and homotopical algebra (among many other things). The original manuscripts of Grothendieck are available online; Pursuing stacks is available on the Grothendieck circle under Unpublished mathematical texts, and here is Les Derivateurs.

There is much one could say about the homotopical ideas of Grothendieck, and I am not really competent enough to say much. There is one small thing that I would like to comment on, which I have found strange since I first saw it, and which even experts, including Grothendieck, refer to as a mystery. It is the appearance of the category in the definition of simplicial sets. A simplicial set is by definition a contravariant functor from  to  (see nLab or Wikipedia). In general, a simplicial object in a category is a contravariant functor from  to .

Somehow an amazing miracle happens when we pass from the category  (extremely boring category, no geometry at all), to the category of such functors into , in which the objects model all homotopy types, so in some sense encompass all the complexity of algebraic topology. A similar thing happens when we pass from the category of abelian groups, which is reasonably boring, to the category of simplicial abelian groups, which correspond, via the Dold-Kan correspondence, to complexes of abelian groups. These form the key tool for extracting cohomological data from geometric objects, and they are the building blocks for the rich world of derived categories. In a similar way one can pass from algebras to simplicial algebras, and get something which is in some sense behaves like the category of dg-algebras – not sure about the details of this though. Yet another example is the shift from schemes to simplicial schemes, which was one of the fundamental ideas in Deligne’s groundbreaking work on mixed Hodge theory in the 70s. In each of these processes, it seems that one passes to a category that is much richer, and which is a much more natural setting for the study of many problems. Why is this?

To make a long story short,  Grothendieck asks where on earth the category  came from. He commences a study of the class of categories which behave like the  category, i.e. which have the amazing property that presheaves on the category model homotopy types, and he calls such a category a test category. More about this in the scanned notes above. 

After Grothendieck, and especially in recent years, quite a lot of work has been done to develop his vision on homotopical algebra and higher categories. We refer to the webpages of Ronnie Brown, Denis-Charles Cisinski, Bertrand Toen, George Maltsinotis and of course to the n-category cafe. Something to look forward to, is that Pursuing stacks is being edited and hopefully soon published, together with much of Grothendieck’s correspondence with various people at the time.

ArXiv finds

Some recent nice preprints (1 Jan – 24 Jan):

Dolgushev, Tamarkin and Tsygan give a really nice introduction to formality theorems, with good references to the original articles by Kontsevich and others.

Dugger and Isaksen: The motivic Adams spectral sequence. Some actual computations!

Wildeshaus: Chow motives without projectivity

Ebert: The homotopy type of a topological stack

Milles: Andre-Quillen cohomology of algebras over an operad.

Bloch: Motivic cohomology

Spencer Bloch gave a nice overview of motivic cohomology. I could not take complete notes, as he was using slides, but fortunately he posted the slides online here.

Toen: Nonabelian Hodge structures

More notes from the Grothendieck conference: Bertrand Toen talked about how one can use derived algebraic geometry to define nonabelian Hodge structures. Here are the scanned notes.

Ofer Gabber and Carlos Simpson

Also on Monday, Carlos Simpson talked about descent, and Ofer Gabber talked about finiteness theorems in etale cohomology. I don’t have very coherent notes from these talks, so I will only point to the abstracts (Gabber and Simpson), both of which are very informative (2 pages each) and much more coherent than anything I could post myself. For Gabber’s rather technical results on finiteness theorems, see also the notes by Joel Riou from a study group in Paris.

I hope to come back to the notion of descent in some future posts, when I have understood it better.

Nick Katz: The l-adic revolution in number theory

On Monday, Nick Katz talked about l-adic representations, in particular the Sato-Tate conjecture. Here are my scanned notes. More notes to come.

Articles on Grothendieck’s life and work

Having returned from a very inspiring week at IHES, I will start going through my notes and post here whatever I think might be useful. To begin with, there are a number of articles on Grothendieck’s life and work, collected on this page. The first one is written by Winfried Scharlau, who is also writing a three-volume biography of Grothendieck (in German), see his web page. Of course, there are many other interesting things on the Grothendieck circle page.

Error in the article of Harada

I wrote earlier about the recent preprints of Harada, in which he claims to prove the standard conjectures of Grothendieck. Having asked some experts what they think about this, most seem sceptical, and my supervisor pointed out to me that there is at least one serious mistake in The Tate-Thomason conjecture. After Theorem 6.3 on page 19, he states that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for varieties of positive characteristics, and he gives a reference to Milne: Polarizations and Grothendieck’s Standard Conjectures. However, Milne only proves that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for abelian varieties, so the argument of Harada is not valid. Of course, if this is the only mistake in the article, his results would still be extremely interesting, but in any case it seems like the standard conjectures are certainly not proven in full.

Going to Paris

After almost a month of illness and/or Christmas holiday, things are now getting back to normal, and tomorrow I’m going to Paris for the Grothendieck conference! I will try to scan and post my notes here when I come back home, in case anyone is interested.