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for equivariant sheaf toposes on open topological groupoids is used to give a proof for the (known, but apparently unpublished) proposition that if $H$ is a subgroupoid of an open topological groupoid $G$, then the topos of equivariant sheaves on $H$ is a subtopos of the topos of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.
Keywords: Grothendieck toposes, sheaves on topological groupoids, categorical logic
2010 MSC: 18B25, 03G30, 18F99
Theory and Applications of Categories, Vol. 28, 2013, No. 18, pp 541-551.