Sites whose topoi are the smooth representations of locally profinite groups

TL;DR

This paper introduces a class of sites whose associated topoi are equivalent to categories of smooth or discrete sets with continuous actions of locally prodiscrete monoids and groups, generalizing known cases like étale fundamental groups.

Contribution

It defines new classes of sites that generalize the relationship between topoi and smooth or discrete representations of locally prodiscrete monoids and groups.

Findings

Topos equivalent to smooth sets of locally prodiscrete monoids

Examples include profinite groups and adelic points of algebraic groups

Generalizes étale fundamental group representations

Abstract

We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined). Examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups. This is a generalization of the fact that the topos associated with the \'etale site of a scheme is equivalent to the category of sets with continuous action by the \'etale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.