We call a finitely complete category diexact if every difunctional relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with finite limits and pushouts of difunctional relations is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos.
Keywords: Exactness, pushouts, difunctional relation
2000 MSC: 18A30, 18B25
Theory and Applications of Categories, Vol. 27, 2012, No. 1, pp 2-9.