On Fibrations Between Internal Groupoids and Their Normalizations

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