The cohomology objects of a semi-abelian variety are small

• Main Authors: Mattenet, Sébastien, Van der Linden, Tim, Jungers, Raphaël M
• Format: Journal Article
• Language: English
• Published: 19.06.2025
• Subjects: Mathematics - Category Theory
• DOI: 10.48550/arxiv.2411.17200

Theory Appl. Categ. 44 (2025), no. 22, 643--663 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).