On actions and strict actions in homological categories

Manfred Hartl and Bruno Loiseau

Let $G$ be an object of a finitely cocomplete homological category $\mathbb C$. We study actions of $G$ on objects $A$ of $\mathbb C$ (defined by Bourn and Janelidze as being algebras over a certain monad $\mathbb T_G$), with two objectives: investigating to which extent actions can be described in terms of smaller data, called action cores; and to single out those abstract action cores which extend to actions corresponding to semi-direct products of $A$ and $G$ (in a non-exact setting, not every action does). This amounts to exhibiting a subcategory of the category of the actions of $G$ on objects $A$ which is equivalent with the category of points in $\mathbb C$ over $G$, and to describing it in terms of action cores. This notion and its study are based on a preliminary investigation of co-smash products, in which cross-effects of functors in a general categorical context turn out to be a useful tool. The co-smash products also allow us to define higher categorical commutators, different from the ones of Huq, which are not generally expressible in terms of nested binary ones. We use strict action cores to show that any normal subobject of an object $E$ (i.e., the equivalence class of $0$ for some equivalence relation on $E$ in $\mathbb C$) admits a strict conjugation action of $E$. If $\mathbb C$ is semi-abelian, we show that for subobjects $X$, $Y$ of some object $A$, $X$ is proper in the supremum of $X$ and $Y$ if and only if $X$ is stable under the restriction to $Y$ of the conjugation action of $A$ on itself. This also amounts to an alternative proof of Bourn and Janelidze's category equivalence between points over $G$ in $\mathbb C$ and actions of $G$ in the semi-abelian context. Finally, we show that the two axioms of an algebra which characterize $G$-actions are equivalent with three others ones, in terms of action cores. These axioms are commutative squares involving only co-smash products. Two of them are associativity type conditions which generalize the usual properties of an action of one group on another, while the third is kind of a higher coherence condition which is a consequence of the other two in the category of groups, but probably not in general. As an application, we characterize abelian action cores, that is, action cores corresponding to Beck modules; here also the coherence condition follows from the others.

Keywords: action, semi-direct product, conjugation, normal subobject, ideal, commutator, homological category, semi-abelian category, algebra over monad

2010 MSC: 18A05, 18A20, 18A22

Theory and Applications of Categories, Vol. 27, 2013, No. 15, pp 347-392.

Published 2013-03-13


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