Australian Category Seminar

Let A be an EM-category and A an object of A. A power object of A in A is an object PA together with a relation $\epsilon_A : PA \to A$ such that for every relation r : X -> A there is a unique morphism f : X -> PA such that $r = \epsilon_A^\circ\Lambda(f)$. If every object has a power object we call A a near-topos. The full higher-order logic of topoi can be interpreted in a near-topos, every PA is complete (proof uses higher-order logic); the complete objects form a reflective subcategory with left exact reflector and the complete objects form an elementary topos. Examples: (i) Any quasitopos, with (E,M) = (Epis, Strong Monos) (ii) Category of topological spaces, with (Epis, Subspace inclusions) (iii) Elementary topos with topology j; E = composites epi with j-dense mono, M = j-closed monos. Here the complete objects are the j-sheaves, so the result above generalizes the sheafification theorem for elementary topoi.