A-homotopy theory of schemes

Publications Mathématiques de l'IHES, 1999

Advances in Mathematics, 2016

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study ofétale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur'sétale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relativeétale homotopy type that was used in [HaSc] for the study of obstructions to the existence of rational points. This relative notion enables to generalize these homotopical obstructions from fields to general base schemas and general maps of topoi. Our model structure is constructed using a general theorem that we prove. Namely, we introduce a much weaker structure than a model category, which we call a "weak fibration category". Our theorem says that a weak fibration category can be "completed" into a full model category structure on its pro-category, provided it satisfies some additional technical requirements. Our model structure is obtained by applying this result to the weak fibration category of simplicial sheaves over a Grothendieck site, where the weak equivalences and the fibrations are local in the sense of Jardine [Jar].

2008

In this article we will study a generalization of the homotopy theory we know from algebraic topology. We discuss the abstract tools needed for this generalization, namely model categories and their homotopy categories. We will apply our general setting to topological spaces to find the familiar homotopy theory.

Advances in Mathematics, 2005

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition ofétale K-theory of ring spectra, extending theétale K-theory of commutative rings.

1998

We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest.

American Journal of Mathematics, 2001

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or 'continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

2006

There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We’re going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subje...

Contemporary Mathematics, 2007

There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and Spalinski [16] is a superb and short introduction, and the books of Hovey [22] and Hirschhorn [21] provide much more in-depth analysis. For a focus on simplicial model categories-model categories enriched over simplicial sets in an appropriate way-one can read [20]. Reaching back a bit further, there's no harm in reading the classics, and Quillen's original monograph [30] certainly falls into that category. Contents 1. Model Categories and Resolutions 2 2. Quillen Functors and Derived Functors 12 3. Generating New Model Categories 17 4. Simplicial Algebras and Resolutions in Non-abelian Settings 21 5. Resolutions in Model Categories 36 References 46

Mathematical Proceedings of the Cambridge Philosophical Society, 1995

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.