Remarks on exactness notions pertaining to pushouts

Richard Garner*

*Corresponding author for this work

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1 Citation (Scopus)
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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 languageEnglish
Pages (from-to)2-9
Number of pages8
JournalTheory and Applications of Categories
Volume27
Publication statusPublished - 20 Mar 2012

Bibliographical note

Copyright the Author(s) [2012]. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

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