Subgroupoids and Quotient Theories

TL;DR

This paper uses Moerdijk's site description to prove a known proposition about subgroupoids and equivariant sheaves, and applies it to characterize subgroupoids definable by quotient theories.

Contribution

It provides an intrinsic characterization of subgroupoids that are definable by quotient geometric theories using topos-theoretic methods.

Findings

Equivariant sheaves on subgroupoids form subtoposes of those on the original groupoid.

The paper offers a new proof of a known proposition using site descriptions.

It characterizes subgroupoids corresponding to quotient theories intrinsically.

Abstract

Moerdijk's site description for equivariant sheaf toposes on open topological groupoids is used to give a proof for the (known, but apparently unpublished) proposition that if H is a strictly full subgroupoid of an open topological groupoid G, then the topos of equivariant sheaves on H is a subtopos of the topos of equivariant sheaves on G. This proposition is then applied to the study of quotient geometric theories and subtoposes. In particular, an intrinsic characterization is given of those subgroupoids that are definable by quotient theories.