Action accessibility via centralizers

Abstract

In this paper we first introduce a non-symmetric notion of centralization between a relation S and an equivalence relation R, which coincides with Smith centralization in the case S is an equivalence relation too. We then prove that in any action accessible category in the sense of Bourn and Janelidze (2009) [11], the centralizer of an equivalence relation R, defined as in [11], actually has a stronger property, namely it is an equivalence relation, which is the largest among all the relations S centralizing R in the non-symmetric sense mentioned above. As a main result, we show that the existence of centralizers for any equivalence relation with this stronger property actually characterizes action accessibility for exact protomodular categories.