The cohomology objects of a semi-abelian variety are small

TL;DR

This paper demonstrates that in semi-abelian varieties, the collection of all n-step extensions between two objects can be represented by a set, ensuring the cohomology objects are 'small' and well-behaved.

Contribution

It proves that the collection of n-step extensions in semi-abelian varieties is essentially a set, addressing a foundational issue in the definition of cohomology objects.

Findings

The collection of all n-step extensions forms a set in semi-abelian varieties.

Extensions can be represented by a set, making cohomology objects 'small'.

Results extend to double, crossed, and Schreier extensions.

Abstract

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).