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Non-additive derived functors via chain resolutions

Part of: Homological algebra

Published online by Cambridge University Press:  28 February 2025

Maxime Culot ,
Fara Renaud  and
Tim Van der Linden Open the ORCID record for Tim Van der Linden [Opens in a new window]
Maxime Culot
Affiliation:
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Fara Renaud
Affiliation:
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Tim Van der Linden*
Affiliation:
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium Mathematics & Data Science, Vrije Universiteit Brussel, 1050 Brussel, Belgium
*
Corresponding author: Tim Van der Linden; Email: tim.vanderlinden@uclouvain.be
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Abstract

Let $F:\; {\mathscr {C}} \to {\mathscr {E}} \ $ be a functor from a category $\mathscr {C} \ $ to a homological (Borceux–Bourn) or semi-abelian (Janelidze–Márki–Tholen) category $\mathscr {E}$. We investigate conditions under which the homology of an object $X$ in $\mathscr {C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathscr {C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case.

Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn–Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathscr {C}$ is a pointed regular category with finite coproducts and enough projectives, provided the class of projectives is closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts—conditions that amount to additivity when $\mathscr {C}$ and $\mathscr {E}$ are abelian categories.

Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.

Keywords

Derived functornormal subtractive categoryhomological categorysemi-abelian categorychain homotopysimplicial homotopyapproximate subtractionprotosplit subobjectprojective object

MSC classification

Primary: 18G05: Projectives and injectives 18G10: Resolutions; derived functors
Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 44
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust.

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