Aspects of algebraic exponentiation

We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category Gp of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category G p, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category. Key words and phrases : Fibration of points, Beck-Chevalley condition, centralizer, unital, Mal'cev and protomodular categories, locally cartesian closed category.