On the \'etale homotopy type of higher stacks

TL;DR

This paper introduces a broad, refined approach to étale homotopy theory applicable to higher stacks and algebraic stacks, establishing a comparison theorem linking étale and topological homotopy types via profinite completion.

Contribution

It develops a new, more general framework for étale homotopy types of higher stacks, extending previous theories to non-Noetherian schemes and providing a profinite comparison theorem.

Findings

The étale homotopy type of higher stacks agrees with the topological homotopy type after profinite completion.

The new approach applies to all algebraic stacks, not just schemes.

A modern reformulation of local systems and cohomology using ∞-categories is provided.

Abstract

A new approach to \'etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the \'etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbitrary higher stack on the \'etale site of affine schemes of finite type over $\mathbb{C},$ then the \'etale homotopy type of $\mathcal{X}$ agrees with the homotopy type of the underlying stack $\mathcal{X}_{top}$ on the topological site, after profinite completion. In particular, if $\mathcal{X}$ is an Artin stack locally of finite type over $\mathbb{C}$, our definition of the \'etale homotopy type of $\mathcal{X}$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\mathcal{X}_{top}$ of $\mathcal{X}$ in the sense of Noohi. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.