#
Remarks on exactness notions pertaining to pushouts

##
Richard Garner

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.

Published 2012-03-20.

http://www.tac.mta.ca/tac/volumes/27/1/27-01.dvi

http://www.tac.mta.ca/tac/volumes/27/1/27-01.ps

http://www.tac.mta.ca/tac/volumes/27/1/27-01.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/1/27-01.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/1/27-01.ps

TAC Home