2012/01/04 by Richard Garner, Garner, Richard
Computer Science · Mathematics · #18A30 #18B25 #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #math.CT #msc:18A30 #msc:18B25
paper · pdf · doi:10.48550/arxiv.1201.0805
7 pages
arxiv created 2012/01/04 · openalex publication_date 2012/01/04 · arxiv updated 2012/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We call a finitely complete category diexact if every Mal'cev 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 Mal'cev spans is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos.