Morphisms and comorphisms of sites II -- Distributors of sites

TL;DR

This paper introduces distributors of sites, generalizing morphisms and comorphisms, and establishes an adjunction linking these distributors to geometric morphisms between sheaf topoi, enriching the categorical understanding of sites.

Contribution

It defines the concept of distributors of sites, explores their properties, and connects them to geometric morphisms via an adjunction, extending the categorical framework of sites.

Findings

Distributors of sites generalize morphisms and comorphisms.

An adjunction between distributors and geometric morphisms is established.

The bicategory of sites and distributors exhibits equipment-like properties.

Abstract

We introduce a notion of distributor of sites, involving suited analogs of flatness and cover-preservation, and show that this notion jointly generalizes those of morphism and comorphism of sites. Given two sites, we exhibit an adjunction between the category of distributors of sites between them and the category of geometric morphisms between the associated sheaf topoi; this adjunction restricts to an equivalence between geometric morphisms and continuous distributors of sites. We finally discuss some equipment-like properties of the bicategory of sites and distributors of sites.