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)»

Here is the second part of my walk-through to Voevodskys A¹-homotopy theory:

On page 48, the first Lemma is shown. Without proof - so I will try to illuminate things a little bit by giving the proof. This lemma isn't used until section 3, so you can skip it, if you want to. I suggest not to do so, if you are intimidated by the diagram, because it isn't that hard, and it's a nice exercise to get the concepts in your head right.

Continue reading «Walk-through to Morel-Voevodsky A¹-homotopy theory part II (page 48, Lemma 1.1)»

As I'm currently reading Morel&Voevodskys paper on A¹-homotopy theory of schemes, I will by and by write a little "walk-through" with hints & comments on how to find additional information to better (or even start to) understand the paper. Maybe for a newcomer to the subject (like me) it's difficult at first to stick together all these concepts like model theory, simplicial objects, Grothendieck topologies, algebraic geometry and so on. I will try to provide helpful comments to make the paper more accessible to those who are not that familiar with these notions above.

Continue reading «Walk-through to Morel-Voevodsky: A¹-homotopy theory of schemes (1999)»