On the exact completion of the classical homotopy category being a pretopos:
On the category of internal categories in a Malcev category:
For the characterisation of categories which are localizations of Maltsev varieties:
On regular categories and protomodular categories:
Dominique Bourn, Marino Gran, Regular, Protomodular, and Abelian Categories, Chapter IV, pp.165-211 in: Maria Pedicchio, Walter Tholen (eds.), Categorical Foundations, Cambridge University Press 2004 (doi:10.1017/CBO9781107340985.007)
Marino Gran, An introduction to regular categories, in: New Perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts, Springer (2021) [arXiv:2004.08964, ISBN:978-3-030-84319-9]
On the Beck-Chevalley condition in Goursat categories:
Presenting a pretorsion theory on Cat whose torsion(-free) objects are the groupoids (skeletal categories, respectively), hence whose “trivial objects” are the skeletal groupoids:
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