Fosco Loregian

We study fibrations arising from indexed categories of the following form: fix two categories A,X and a functor F : A x X -> X , so that to each F_a=F(a,_) one can associate a category of algebras Alg_X(F_a) (or an Eilenberg-Moore, or a Kleisli category if each $F_a$ is a monad). 

We call the functor A ⧔ Alg -> A, whose typical fibre over A is the category Alg_X(F_a), the <<fibration of algebras>> obtained from F.

Examples of such constructions arise in disparate areas of mathematics, and are unified by the intuition that A ⧔ Alg  is a form of <<semidirect product>> of the category A, acting on X, via the "representation" given by the functor F: A x X -> X.

A motivating example lies in the theory of Hopf algebras: there is an action of the category o groups, via the group algebra functor, on Lie algebras, and the semidirect product recognizes the category of cocommutative Hopf algebras on a char 0 field.