Toposes Over A Base

Abstract We are interested in this chapter in studying the 2-category Top/S of toposes over a fixed topos S, through the medium of indexed category theory. We recall the precise definition of this 2-category from Section A4.1: its objects (which we call toposes defined over S, or simply S-toposes) are geometric morphisms p: E — S with codomain S, its morphisms (q: F — S) — (p: E — S) are pairs (f, a) where f: F — E is a geometric morphism and a : q = pf a geometric transformation, and its 2-cells (f, a) — (g, fi) are geometric transformations f — g compatible in the obvious sense with a and fi. However, we shall almost invariably abuse notation by suppressing any mention of the 2-isomorphism a when specifying 1-cells of Top/S; we shall also tend to suppress the structural morphism p when specifying objects of Top/S, and simply write ‘E is an S-topos’. (The justification for the latter abuse is contained in Theorem 3.1.2 below.)