Left exact presheaves on a small pretopos Original Research Article

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