Two Ways to Centralizers of Equivalence Relations

Recent works dealing with new considerations about the notion centralizer of equivalence relations gave the opportunity to reveal beyond this construction widerranging phenomenons of functorial nature which do not belong, as it is well-known, to this notion by itself. And this was done in two distinct ways: on the one hand through the notion of action accessible category [2] and on the other hand through the notion of fibrewise algebraically cartesian closed (facc) category [1] which deals with the existence of centralizers of subobjects in the fibres PtY E of the fibration points: ¶E : PtE → E. This two notions are both illustrated by the categories Gp of groups and R-Lie of Lie R-algebras. The aim of this talk is to elucidate what is their common core and to analyse how they are articulated in general.