Partial Toposes Jean Bénabou and Thomas Streicher

We introduce various notions of partial topos, i.e. “topos without terminal object”. The strongest one, called local topos, is motivated by the key examples of finite trees and sheaves with compact support. Local toposes satisfy all the usual exactness properties of toposes but are neither cartesian closed nor have a subobject classifier. Examples for the weaker notions are local homeomorphisms and discrete fibrations. Finally, for partial toposes with supports we show how they can be completed to toposes via an inverse limit construction. 1. Partial Toposes For a category B with pullbacks its fundamental fibration PB = ∂1 : B 2 → B is well– powered [Bénabou, 1980] iff for every a : A → I in B/I there is an object p(a) : P (a) → I in B/I together with a subobject a : Sa P (a)×IA such that for every b : B → I and subobject m of B×IA there is a unique map χ : b → p(a) with m ∼= (χ×IA)∗ a as in the diagram