The aim of the talk is two-fold:
(i) to explain that finitely co-complete pretoposes are co-ideally-exact [J] and co-arithmetical [P], and (ii) to explain that there are conditions in common between additive and lextensive categories, which together with exactness (and finite cocompleteness) imply co-exactness and co-protomodularity. As a consequence of (i), we recover Theorem 1.3 of [BC], which states that the dual of the category of pointed compact Hausdorff spaces is semi-abelian, which in fact was the motivation to begin this work.
References: [BC] F. Borceux, M.M. Clementino, On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces, Theory and Applications of Categories 43(11), 363-381, 2025. [J] G. Janelidze, Ideally exact categories, Theory and Applications of Categories 41(11), 414-425, 2024. [P] M.C. Pedicchio, A categorical approach to commutator theory, Journal of Algebra 177, 647-657, 1995.
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