Abstract
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.
Original language | English |
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Pages (from-to) | 2-9 |
Number of pages | 8 |
Journal | Theory and Applications of Categories |
Volume | 27 |
Publication status | Published - 20 Mar 2012 |