The cohomology objects of a semi-abelian variety are small

A well-known, but often ignored issue in Yoneda-style definitions of cohomology objects via collections of n-step extensions (i.e., equivalence classes of exact sequences of a given length n between two given objects, usually subject to further criteria, and equipped with some algebraic structure) is, whether such a collection of extensions forms a set. We explain that in the context of a semi-abelian variety of algebras, the answer to this question is, essentially, yes: for the collection of all n-step extensions between any two objects, a set of representing extensions can be chosen, so that the collection of extensions is "small" in the sense that a bijection to a set exists. We further consider some variations on this result, involving double extensions and crossed extensions (in the context of a semi-abelian variety), and Schreier extensions (in the category of monoids).

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Mattenet, S., Van der Linden, T., & Jungers, R. (2025). The cohomology objects of a semi-abelian variety are small. Theory and Applications of Categories, 44(22), 643-663. https://doi.org/10.48550/arXiv.2411.17200 (Original work published 2025)