Related papers: Nonabelian $H^1$ and the \'Etale van Kampen Theore…

We study a condensed version of the \'etale homotopy type of a scheme, which refines both the usual \'etale homotopy type of Friedlander-Artin-Mazur and the pro\'etale fundamental group of Bhatt-Scholze. In the first part of this paper, we…

We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic…

We show that functors like algebraic $K$-theory (such as unitary or symplectic $K$-functors), as well as the higher Grothendieck--Witt groups, possess the local constancy condition for Henselian valuation rings. Namely, taken with finite…

This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in \'etale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number…

We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations…

This paper is a survey on the homotopy theory of $E_n$-operads written for the new handbook of homotopy theory.

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This…

We extend the classical (connected, etale) factorization of locally connected geometric morphisms into a (terminally connected, pro-etale) factorization for all geometric morphisms between Grothendieck topoi. We discuss properties of both…

We develop analysis of H\"older continuous mappings with applications to geometry and topology of the Heisenberg groups. We cover the theory of distributional Jacobians of H\"older continuous mappings and pullbacks of differential forms…

We prove that every Grothendieck topology induces a hereditary torsion pair in the category of presheaves of modules on a ringed site, and obtain a homological characterization of sheaves of modules: a presheaf of modules is a sheaf of…

This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally finite nilpotent homotopy can be represented by a simplicial set which is a finitely generated free group in…

For the Galois closure $\Xgal$ of a generic projection from a surface $X$, it is believed that $\pi_1(\Xgal)$ gives rise to new invariants of $X$. However, in all examples this group is surprisingly simple. In this article, we offer an…

We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in…

The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the $\infty$-categorical approach, as developed by Lurie. Three…

This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois…

We study a non-pointed version of the notion of torsion theory in the framework of categories equipped with a posetal monocoreflective subcategory such that the coreflector inverts monomorphisms. We explore the connections of such torsion…

We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmueller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of…

We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the…

Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show…

Let $\mathbb{D}$ be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site $X$, there exists a reflector from the category of precosheaves on $X$ with values in $\mathbb{D}$ to the full subcategory of…