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 stru...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Mattenet, Sébastien, Van der Linden, Tim, Jungers, Raphaël M
Format: Article
Language:English
Subjects:
Algebra
Collection
Context
Homology
Monoids
Online Access:Get full text
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Summary: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).
ISSN:2331-8422