Grothendieck quantaloids for allegories of enriched categories

Abstract

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