Allegories of enriched categories
For any small involutive quantaloid Q, let SymDist(Q_ssi) be the (large) involutive quantaloid whose objects are the symmetric categories enriched in the split-symmetric-idempotent completion of Q, and morphisms are distributors. I shall characterise those quantaloids Q, called Grothendieck quantaloids, for which SymDist(Q_ssi) is the category of relations in a topos. That topos, written Sh(Q), is then recovered by taking left adjoints in SymDist(Q_ssi). Examples include locales, sites, and étale groupoids. Joint work with Hans Heymans.