39 research outputs found
A note on strong protomodularity, actions and quotients
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
Bourn-normal monomorphisms in regular Mal'tsev categories
Normal monomorphisms in the sense of Bourn describe the equivalence classes
of an internal equivalence relation. Although the definition is given in the
fairly general setting of a category with finite limits, later investigations
on this subject often focus on protomodular settings, where normality becomes a
property. This paper clarifies the connections between internal equivalence
relations and Bourn-normal monomorphisms in regular Mal'tesv categories with
pushouts of split monomorphisms along arbitrary morphisms, whereas a full
description is achieved for quasi-pointed regular Mal'tsev categories with
pushouts of split monomorphisms along arbitrary morphisms.Comment: This vesion fixes one error present in the last section of the
previous versio
Braided and symmetric internal groupoids
Braided and symmetric internal groupoids in semi-abelian categories are discussed
A note on strong protomodularity, actions and quotients
We investigate some properties of the fibration of points. We obtain a
characterization of protomodular categories among pointed regular ones,
and, in the semi-abelian case, a characterization of strong protomodularity. Everything is also stated in terms of internal actions
The snail lemma for internal groupoids
We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizer
A push forward construction and the comprehensive factorization for internal crossed modules
In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids