The Pro-Étale Homotopy Type
Paul Meffle
TL;DR
The paper introduces the pro-étale homotopy type $\Pi_{pe}(S)$ for schemes, extending Artin–Mazur’s étale homotopy theory to the pro-étale site. It proves that for qcqs schemes this type is profinite and computable from a single split affine weakly contractible hypercovering, and it develops cohomology via $\uppi$-sheaves that recovers pro-étale cohomology. It also computes notable examples (e.g., $\mathbb{R}$) and provides a refined pro-étale homotopy type using condensed sets and Barnea–Schlank’s framework, enabling definitions of pro-étale homotopy groups. The work situates these constructions within existing literature, establishes (and partially validates) the expected compatibility with space of components, and lays groundwork for understanding the pro-étale homotopy type of fields. Overall, it offers a concrete, model-category–friendly approach to pro-étale homotopy theory with concrete computational tools and refined invariants.
Abstract
In this paper we define the pro-étale homotopy type of a scheme and prove some of its expected properties. Our definition is similar to the definition of the étale homotopy type by Michael Artin and Barry Mazur. We prove that for a qcqs scheme the pro-étale homotopy type is profinite, determined by a single split affine weakly contractible hypercovering and computes the cohomology of a certain class of sheaves. We show that the pro-étale homotopy type of a w-contractible scheme is trivial and compute the pro-étale homotopy type of the real numbers. Moreover, we prove that a suitable version of $π_0$ composed with the pro-étale homotopy type gives back the space of components of the base scheme. We make some progress towards describing the pro-étale homotopy type of arbitrary fields. Lastly, we give a refined definition of the pro-étale homotopy type using the theory by Ilan Barnea and Tomer M. Schlank and the theory of condensed sets by Dustin Clausen and Peter Scholze. This allows us to define pro-étale homotopy groups associated to pointed qcqs schemes.