A note on strong protomodularity, actions and quotients

Abstract

In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions. IntroductionThe present work originates from the investigation of the categorical properties related to two well-known features of group actions. Actions on quotientsSuppose we are given a pair (ξ, g):where ξ is a left-action of groups, and g is a surjective homomorphism. We discuss the following problem: under what conditions does the action ξ induces an action on the quotient Z? Indeed, it is not difficult to see that ξ is well-defined on the cosets of Y mod X = Ker(g), precisely when it is well-defined on the 0-coset X, i.e. when it restricts to X. We shall state this property as follows:(KC) An action passes to the quotient if, and only if, it restricts to the kernel. Action of quotientsSuppose now that we are given a group action ξ as before, and a surjective group homomorphism q : A → Q. A natural question arises: when does the given A-action induce a Q-action? In this case, the restriction of the action ξ to the kernel K of q always exists, and the condition under which the action of the quotient is well defined, amounts to the fact that the kernel of q acts trivially. 1These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category (see [10]). Indeed, we will show that the property (KC) characterizes strongly protomodular categories among semi-abelian categories, and that, in such contexts, actions of quotients behave substantially in the same way as in the case of groups.On the other hand these issues can be dealt with also in more general contexts. Indeed, when an object A acts on object X, just like in the case of group, one can consider the split epimorphism X ⋊ A → A given by the semidirect product projection together with its canonical section. Vice-versa, any split epimorphism with codomain A gives rise to the conjugation A-action on the kernel of the split epimorphism.This allows to formulate our issues in terms of split epimorphisms, or points, even in contexts where the machinery of internal actions is not at all available. This line of investigation will lead us to the study of some new classifying aspects of the fibration of points. In particular, with Proposition 3.3, we will give a characterization of protomodular categories among quasi-pointed regular ones as those with kernel functors that reflect short exact sequences. Then, we will show that the problem of extending actions along quotients translates (in term of points) in a property closely connected with strong protomodularity, i.e. the fact that kernel functors reflect kernels In fact, th...

Compatible actions of Lie algebras

“…We prove it for the first one since the reasoning can be repeated for the other one: For the second part of the claim it suffices to apply Theorem 5.5 in [17] and use the fact that, as shown in [15], Lie R is a strongly protomodular category in the sense of [2].…”

Peiffer product and Peiffer commutator for internal pre-crossed modules

“…In a semi-abelian category C which is strongly protomodular the above condition on the B-action comes for free, so it suffices to ask that k is a kernel. In fact, in the semi-abelian context, the condition "every action which restricts to a kernel passes to the quotient" is equivalent to strong protomodularity (see [23] for details).…”

Fibered aspects of Yoneda's regular span

“…A large part of the theory of crossed modules of groups can be carried on internally in the strongly semi-abelian context (see [39,35]). For instance, one can prove that the kernel B of ∂ is abelian (in fact, central in G 2 ) and that the cokernel C of ∂ acts internally on B, so that ∂ induces on B a structure of C-module.…”

“…So far, we have proved that β, −j is equivariant with respect to the G 1actions. Hence, by Theorem 5.5 in [35], the quotient is also equivariant and B ′ × B G 2 is endowed with a G 1 -action, that we denote by ξ ′ . We are going to see that this action gives rise to a crossed module structure on ∂ ′ .…”