Holmstrom Etale homotopy theory

Something by Ambrus Pal?

Pal conversation (Sep 2011). There is an absolute and a relative homotopy type. The relative one comes with a Galois action. These things are objects in Pro-Ho(sSet) and Pro-Ho(G-sSet) respectively. For the homotopy theory of the latter, see a paper of Goerss (profinite gps…). There is an A1-invariance theorem in Friedlander somewhere, at least for the absolute. One can also work with Pro-objects in the model cat and not the homotopy cat, but then one has to work with rigidified hypercoverings, i.e. geometrically pointed hypercoverings.

Pal said a hypercovering is a simplicial object in the cat of coverings, which is Kan contractible, i.e. trivially fibrant.

arXiv:1110.0164 Homotopy Obstructions to Rational Points from arXiv Front: math.AT by Yonatan Harpaz, Tomer M. Schlank In this paper we propose to use a relative variant of the notion of the étale homotopy type of an algebraic variety in order to study the existence of rational points on it. In particular, we use an appropriate notion of homotopy fixed points in order to construct obstructions to the local-global principle. The main results in this paper are the connections between these obstructions and the classical obstructions, such as the Brauer-Manin, the étale-Brauer and certain descent obstructions. These connections allow one to understand the various classical obstructions in a unified framework.

Artin and Mazur LNM0100. Ass to any locally noetherian prescheme a pro-object in the homotopy cat of simplicial sets, called the etale homotopy type. For a complex variety, this is a certain profinite completion of the usual homotopy type. We develop homotopy theory for pro-simplicial sets, Hurewicz, Whitehead, Postnikoff etc. Relation to questions about independence of embedding. Finiteness of the number of (stable) homotopy types with isomorphic profinite completion. Galois action on the l-adic homotopy type of a variety - this contains more info than cohomology if the var is not abelian. Actual def of etale homotopy type: Take C a locally connected site. Let HR(C) be the cat of hypercoverings of C. Apply the functor pi to the category gives a pro-object in the homotopy cat of simplicial sets. This gives the Verdier functor from locally conn sites to such pro-objects. Here the functor pi is the “connected components functor” from C to Sets. So the “index cat” for the pro-simplicial set is a cat of hypercoverings. Take C to be the etale site over X. Get pro-homotopy and pro-homology groups of C, and also cohomology etc.

Toen: Segal topoi and stacks over Segal categories. File Toen web prepr msri.pdf. Analogous and in some sense equivalent to the notion of model topos, but some advantages. Includes a Giraud type statement characterizing Segal topoi among Segal cats. Notion of homotopy type of a Segal site, improving on the etale homotopy theory of schemes, and more generally on the theory of homotopy types of sites as defined by Artin and Mazur.

arXiv:1002.1423 The Étale Homotopy Type and Obstructions to the Local-Global Principle from arXiv Front: math.AT by Yonatan Harpaz, Tomer M. Schlank In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety [AMa69]. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those “homotopy obstructions” and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

arXiv:1109.5477 A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type from arXiv Front: math.AG by Ilan Barnea, Tomer M. Schlank In [Isa], Isaksen showed that a proper model category $\cC$, induces a model structure on the pro category $Pro(\cC)$. In this paper we generalize Isaksen’s theorem to the case when $\cC$ possess a weaker structure, which we call a “weak fibration category”. Namely, we show that if $\cC$ is a weak fibration category, that satisfies an extra condition, there is a naturally induced model structure on $Pro(\cC)$. We then apply our theorem to the case when $\cC$ is the weak fibration category of simplicial sheafs on a Grothendieck site, where both weak equivalences and fibrations are local as in [Jar]. This gives a new model structure on the category of pro simplicial sheaves. Using this new model structure we give a definition of the étale homotopy type of Artin and Mazur [AM], as the result of applying a derived functor. Our definition actually gives as object in $Pro(\cS)$ and not just in $Pro(Ho(\cS))$ as in [AM]. Our definition also extends naturally to a relative notion of the étale homotopy type, as considered for example in [HaSc].

arXiv:1002.3532 Etale Homotopy Types and Bisimplicial Hypercovers from arXiv Front: math.AT by Michael D. Misamore An étale homotopy type $T(X, z)$ associated to any pointed locally fibrant connected simplicial sheaf $(X, z)$ on a pointed locally connected small Grothendieck site $(\mc{C}, x)$ is studied. It is shown that this type $T(X, z)$ specializes to the étale homotopy type of Artin-Mazur for pointed connected schemes $X$, that it is invariant up to pro-isomorphism under pointed local weak equivalences (but see [Schmidt1] for an earlier proof), and that it recovers abelian and nonabelian sheaf cohomology of $X$ with constant coefficients. This type $T(X, z)$ is compared to the étale homotopy type $T_b(X, z)$ constructed by means of diagonals of pointed bisimplicial hypercovers of $x = (X, z)$ in terms of the associated categories of cocycles, and it is shown that there are bijections \pi_0 H_{\hyp}(x, y) \cong \pi_0 H_{\bihyp}(x, y) at the level of path components for any locally fibrant target object $y$. This quickly leads to natural pro-isomorphisms $T(X, z) \cong T_b(X, z)$ in $\Ho{\sSet_\ast}$. By consequence one immediately establishes the fact that $T_b(X, z)$ is invariant up to pro-isomorphism under pointed local weak equivalences. Analogous statements for the unpointed versions of these types also follow.

arXiv:1002.3530 Nonabelian $H^1$ and the Étale van Kampen Theorem from arXiv Front: math.AG by Michael D. Misamore Generalized étale homotopy pro-groups $\pi_1^{\ets}(\mc{C}, x)$ associated to pointed connected small Grothendieck sites $(\mc{C}, x)$ are defined and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained. Applications include new rigorous proofs of some folklore results around $\pi_1^{\ets}(\et{X}, x)$, a description of Grothendieck’s short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem which gives a simple statement about a pushout square of pro-groups that works for covering families which do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck’s profinite groups $\pi_1^{\Gals}$ immediately follows.

nLab page on Etale homotopy theory?