Geodesic
Remarks on exactness notions pertaining to pushouts
Theory and applications of categories, CT2011, Tome 27 (2012), pp. 2-9.

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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.
Publié le :
Classification : 18A30, 18B25
Keywords: Exactness, pushouts, difunctional relation 
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Richard Garner. Remarks on exactness notions pertaining to pushouts. Theory and applications of categories, CT2011, Tome 27 (2012), pp. 2-9. http://geodesic.mathdoc.fr/item/TAC_2012_27_a0/