This is a very short notice to memorize some of the various model structures on simplicial presheaves in a systematic way.

[UPDATE 2013-03-06] I gave a talk in our working group seminar about model structures on simplicial presheaves, homotopy sheaves and h-principles [/UPDATE]

Continue reading «Model structures on simplicial presheaves»

Last week Matthias Wendt and I have put our first joint paper on the arXiv, it's called "A¹-fundamental groups of isotropic reductive groups" and here I will tell you how you can think about it without going through all definitions.

I have blogged about this before, but until a few days ago, there was only my diploma thesis in german and an extended abstract in english online, and the results were not as general (oh, and the proofs were also not the best...).

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These days I read Akhil Mathew's post on Vaserstein's proof of the Quillen-Suslin theorem, once known as Serre's Problem. This inspired the following.

Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

Continue reading «Is it possible to prove Serre's Problem (the Quillen-Suslin theorem) via Motivic Homotopy ...»

Nearly half of all blog posts start with "Soon this blog will be full of content" and the other half with "it has been incredibly silent, soon I will post a lot". This post is of the second type, but I apologize by linking to what has kept me from writing here:
Extended abstract of my diploma thesis.
While the diploma thesis is in german, the abstract is in english and only 9 pages long, without any proofs. The diploma thesis is now available here (updated on 2011-08-09).

As a teaser, here is the abstract of the abstract:

In classical covering space theory we have an isomorphism of the fundamental group with the fibre of the universal cover over the basepoint. Covering spaces of topological groups are group extensions, but not every group extension is a covering space. Perfect groups admit a universal central extension and the kernel of this extension is also called fundamental group. For simply connected Chevalley-groups over a perfect field, this fundamental group, classically called second unstable K-Theory, is exactly the fundamental group of a simplicial resolution. The loops are described explicitly by matrices.

We look at the model structure Voevodsky and Morel use in their 1999 IHES paper and discuss 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.10. There is nothing difficult or particularly interesting, but you might want to look up some specific issue or reference.
Continue reading «Walk-through to Morel-Voevodsky A¹-homotopy theory, page 48-50»

I decided to post some background needed in order to understand Morel-Voevodsky's paper "A¹-homotopy theory". I explain some notions of simplicial sets, topoi, monoidal categories, enriched categories and simplicial model categories.

I tried to give many more references I found useful.
Continue reading «Categorical background for A¹-homotopy theory (simplicial model categories)»