Action accessibility via centralizers

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. © 2012 Elsevier B.V..