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manuscripts of Grothendieck are available online; Pursuing stacks is available on the Grothendieck circle under Unpublished mathematical texts, and here is Les Derivateurs.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.
