We propose a new approach to S-protomodular categories in the sense of D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral. Instead of points (=split epimorphisms) it uses generalized points, which we define as composable pairs of morphisms whose composites are pullback stable regular epimorphisms. This approach is convenient in describing the connection between split and regular Schreier epimorphisms of monoids.
Keywords: generalized points, strong generalized points, S-protomodular categories, T-protomodular categories, regular Schreier extensions, split Schreir extensions, monoids
2020 MSC: 18E13, 18A20
Theory and Applications of Categories, Vol. 37, 2021, No. 14, pp 378-387.