Search results for "protomodular"
showing 3 items of 3 documents
On Fibrations Between Internal Groupoids and Their Normalizations
2018
We characterize fibrations and $$*$$ -fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for $$*$$ -fibrations from the internal version of the Gabriel–Zisman exact sequence. We also analyse fibrations and $$*$$ -fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma.
A NOTE ON THE CATEGORICAL NOTIONS OF NORMAL SUBOBJECT AND OF EQUIVALENCE CLASS
2021
In a non-pointed category E, a subobject which is normal to an equivalence relation is not necessarily an equivalence class. We elaborate this categorical distinction, with a special attention to the Mal'tsev context. Moreover, we introduce the notion of fibrant equipment, and we use it to establish some conditions ensuring the uniqueness of an equivalence relation to which a given subobject is normal, and to give a description of such a relation.
A note on strong protomodularity, actions and quotients
2013
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.