For a relative exact homological category (C,E), we define relative points over an arbitrary object in C, and show that they form an exact homological category. In particular, it follows that the full subcategory of nilpotent objects in an exact homological category is an exact homological category. These nilpotent objects are defined with respect to a Birkhoff subcategory in C as defined by T. Everaert and T. Van der Linden. In addition, we introduce relative internal actions and show that, just as in the classical case, there is an equivalence of categories between the category of relative points over an object and the category of relative internal actions for the same object.
Keywords: relative semi-abelian category, relative homological category, semi-abelian category, homological category, category of relative points, relative internal actions
2010 MSC: 18G50, 18G25, 18A20, 18A05, 18A25, 18A32, 18C20
Theory and Applications of Categories, Vol. 31, 2016, No. 16, pp 444-461.