Fibrations of algebras

We study fibrations arising from indexed categories of the following form: fix two categories A, X and a functor F: A X, so that to each FA=F (A, -) one can associate a category of algebras AlgX (FA) (or an Eilenberg-Moore, or a Kleisli category if each FA is a monad). We call the functor ^AAlg A, whose typical fibre over A is the category AlgX (FA), the "fibration of algebras" obtained from F. Examples of such constructions arise in disparate areas of mathematics, and are unified by the intuition that AAlg is a form of semidirect product of the category A, acting on X, via the `representation' given by the functor F: A X. After presenting a range of examples and motivating said intuition, the present work focuses on comparing a generic fibration with a fibration of algebras: we prove that if A has an initial object, under very mild assumptions on a fibration p: E A, we can define a canonical action of A letting it act on the fibre E_ over the initial object. This result bears some resemblance to the well-known fact that the fundamental group ₁ (B) of a base space acts naturally on the fibers Fb = p^-1b of a fibration p: E B.