Left exact presheaves on a small pretopos

Abstract

Given a small exact category E with finite colimits, we prove that the category Lex( E ) of left exact presheaves on E is exact precisely when in E , the equivalence relation generated by a reflexive symmetric relation R is a finite iterate of R. This is in particular the case when E is Noetherian, that is, every ascending chain of subobjects is stationary. When this condition is satisfied and moreover E is a pretopos, Lex( E ) becomes a topos. Various examples are given, distinguishing the possible situations.