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).
Keywords: Yoneda extension; double extension; crossed extension; Schreier extension; cohomology; semi-abelian category; set
2020 MSC: 03E25, 03E30, 18E13, 18G15, 18G50
Theory and Applications of Categories, Vol. 44, 2025, No. 22, pp 643-663.
Published 2025-07-24.
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