Higher Galois theory

TL;DR

This paper extends Galois theory to higher topoi, establishing equivalences between locally constant sheaves and representations of fundamental pro-groupoids, and relating these to classical constructions.

Contribution

It generalizes Galois theory to higher topoi, connecting locally constant sheaves with fundamental pro-groupoids and Galois torsors in a novel way.

Findings

Locally constant sheaves in an (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid.

Finite locally constant sheaves in an infinity-topos correspond to finite representations of its fundamental pro-infinity-groupoid.

The fundamental pro-infinity-groupoid of 1-topoi relates to Artin-Mazur and Friedlander's constructions.

Abstract

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 terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the \'etale topos of a scheme, to its refinement by Friedlander.