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 π₁(B) of a base space acts naturally on the fibers Fb = p⁻¹b of a fibration p : E → B.