Grothendieck quantaloids for allegories of enriched categories

TL;DR

This paper introduces Grothendieck quantaloids, a class of involutive quantaloids, and explores their relationship with sheaves, relations, and toposes, extending the theory of enriched categories and sheaf topos constructions.

Contribution

It defines Grothendieck quantaloids via conditions on Q, linking them to topos theory and enriched categories, and characterizes when categories of Q-sheaves form Grothendieck toposes.

Findings

Rel(Q) is a topos if Q is a Grothendieck quantaloid.

Sh(Q) is a Grothendieck topos for Grothendieck quantaloids.

Examples include locales, closed cribles, and inverse quantal frames.

Abstract

For any small involutive quantaloid Q we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that Rel(Q) is the category of relations in a topos if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; in this case we call Q a Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever Q is a Grothendieck quantaloid. Any locale L is a Grothendieck quantale, and Sh(L) is the topos of sheaves on L. Any small quantaloid of closed cribles is a Grothendieck quantaloid, and if Q is the quantaloid of closed cribles in a Grothendieck site (C,J) then Sh(Q) is equivalent to the topos Sh(C,J). Any inverse quantal frame is a Grothendieck quantale, and if O(G) is the inverse quantal frame naturally associated with an \'etale groupoid G then Sh(O(G)) is the classifying topos of G.