Etale realization on the-homotopy theory of schemes

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.

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.

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.

Topology and its Applications, 2007

We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.

This paper aims to help the development of new models of homotopy type theory, in particular with models that are based on realizability toposes. For this purpose it develops the foundations of an internal simplicial homotopy that does not rely on classical principles that are not valid in realizability toposes and related categories.

Geometry & Topology, 2019

We give a tool for understanding simplicial desuspension in A 1-algebraic topology: we show that X → Ω(S 1 ∧ X) → Ω(S 1 ∧ X ∧ X) is a fiber sequence up to homotopy in 2-localized A 1 algebraic topology for X = (S 1) m ∧ G ∧q m with m > 1. It follows that there is an EHP sequence spectral sequence Z (2) ⊗ π A 1 n+1+i (S 2n+2m+1 ∧ (Gm) ∧2q) ⇒ Z (2) ⊗ π A 1 ,s i (S m ∧ (Gm) ∧q).

Geometry & Topology, 2018

We establish a relative version of the abstract "affine representability" theorem in A 1-homotopy theory from part I of this paper. We then prove some A 1-invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in A 1-homotopy theory. 14F42, 14L10, 20G15, 55R15 1 Introduction Suppose k is a fixed commutative unital base ring, and write H .k/ for the Morel-Voevodsky A 1-homotopy category over k [45]. The category H .k/ is constructed as a certain localization of the category of simplicial presheaves on Sm k , the category of smooth k-schemes. Write Sm aff k for the subcategory of Sm k consisting of affine schemes. If X is a simplicial presheaf on Sm k , by an "affine representability" result for X , we will mean, roughly, a description of the presheaf on Sm aff k defined by U 7 ! OEU; X A 1 WD Hom H .k/ .U; X /.

Journal of Pure and Applied Algebra, 2002

In this paper, we discuss in detail a site for simplicial spaces which is particularly suitable for deÿning derived functors for maps between simplicial spaces. It is shown that the derived category of sheaves on this site is closely related to the derived category of sheaves on another well-known site. Applications to algebraic group actions in positive characteristics are also discussed.

1998

This work was motivated in part by the following question of Soulé: given a simplicial presheaf X on a site C, how does one produce a map of simplicial presheaves X → L HZ X in such a way that each of the maps in sections X(U ) → L HZ X(U ), U ∈ C, is an integral homology localization map in the sense of Bousfield? Secondly, if Y is a simplicial presheaf which is integrally homology local in a suitable sense, is it the case that the map X → L HZ X induces an isomorphism

2022

This dissertation is concerned with the foundations of homotopy theory following the ideas of the manuscripts Les Dérivateurs and Pursuing Stacks of Grothendieck. In particular, we discuss how the formalism of derivators allows us to think about homotopy types intrinsically, or, even as a primitive concept for mathematics, for which sets are a particular case. We show how category theory is naturally extended to homotopical algebra, understood here as the formalism of derivators. Then, we proof in details a theorem of Heller and Cisinski, characterizing the category of homotopy types with a suitable universal property in the language of derivators, which extends the Yoneda universal property of the category of sets with respect to the cocomplete categories. From this result, we propose a synthetic re-definition of the category of homotopy types. This establishes a mathematical conceptual explanation for the the links between homotopy type theory, ∞-categories and homotopical algebra, and also for the recent program of re-foundations of mathematics via homotopy type theory envisioned by Voevodsky. In this sense, the research on foundations of homotopy theory reflects in a discussion about the re-foundations of mathematics. We also expose the theory of Grothendieck-Maltsiniotis ∞-groupoids and the famous Homotopy Hypothesis conjectured by Grothendieck, which affirms the (homotopical) equivalence between spaces and ∞-groupoids. This conjectured, if proved, provides a strictly algebraic picture of spaces.

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.

arXiv: Algebraic Topology, 2019

We undertake a systematic study of the notion of fibration in the setting of abstract simplicial complexes, where the concept of `homotopy' has been replaced by that of `contiguity'. Then a fibration will be a simplicial map satisfying the `contiguity lifting property'. This definition turns out to be equivalent to a known notion introduced by G. Minian, established in terms of a cylinder construction $K \times I_m$. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration have the same strong homotopy type, a notion that has been recently introduced by Barmak and Minian; any fibration with a strongly collapsible base is fibrewise trivial; and some other ones. We introduce the concept of `simplicial finite-fibration', that is, a map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration $PK \...

2003

In [To-Ve2] we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of S-site and of model site, and the associated categories of stacks on them. This led us to a notion of model topos. In this paper we treat the analogous theory starting from (1-)Segal categories in place of S-categories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over them, giving rise to a definition of Segal topos. We compare the notions of Segal topoi and of model topoi, showing that the two theories are equivalent in some sense. However, the existence of a nice Segal category of morphisms between Segal categories allows us to improve the treatment of topoi in this context. In particular we construct the 2-Segal category of Segal topoi and geometric morphisms, and we provide a Giraud-like statement charaterizing Segal topoi among Segal categories. As example of applications, we show how to reconstruct a topological space from the Segal topos of locally constant stacks on it, thus extending the main theorem of [To] to the case of un-based spaces. We also give some hints of how to define homotopy types of Segal sites: this approach gives a new point of view and some improvements on theétale homotopy theory of schemes, and more generally on the theory of homotopy types of Grothendieck sites as defined by Artin and Mazur.

Homology, Homotopy and Applications, 2007

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in Ch. After an appropriate tensor and cotensor structure in Ch is established (§1, §3), Quillen's axiom SM7 is verified in §5 and §6. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing Ch with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) (§7). In §8 the model structure on Ch is shown to be proper.

Applied Categorical Structures, 2004

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity' while an exterior map is a continuous map which is 'continuous at infinity'. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T, T } of suitable exterior spaces. For this goal, for a given exterior space T , we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown-Grossman,Čerin-Steenrod, p-cylindrical, Baues-Quintero and Farrell-Taylor-Wagoner groups, as well as the standard (Hurewicz) homotopy groups. The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T, T } , it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.