Summary: We study fibrations arising from indexed categories of the following form: fix two categories \(\mathcal{A},\mathcal{X}\) and a functor \(F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X} \), so that to each \(F_A=F(A,-)\) one can associate a category of algebras \(\mathbf{Alg}_\mathcal{X}(F_A)\) (or an Eilenberg-Moore, or a Kleisli category if each \(F_A\) is a monad). We call the functor \(\int^{\mathcal{A}}\mathbf{Alg} \to \mathcal{A}\), whose typical fibre over \(A\) is the category \(\mathbf{Alg}_\mathcal{X}(F_A)\), the ”fibration of algebras” obtained from \(F\). Examples of such constructions arise in disparate areas of mathematics, and are unified by the intuition that \(\int^\mathcal{A}\mathbf{Alg} \) is a form of semidirect product of the category \(\mathcal{A}\), acting on \(\mathcal{X}\), via the ‘representation’ given by the functor \(F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X}\). 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 \(\mathcal{A}\) has an initial object, under very mild assumptions on a fibration \(p : \mathcal{E}\longrightarrow \mathcal{A}\), we can define a canonical action of \(\mathcal{A}\) letting it act on the fibre \(\mathcal{E}_\varnothing\) over the initial object. This result bears some resemblance to the well-known fact that the fundamental group \(\pi_1(B)\) of a base space acts naturally on the fibers \(F_b = p^{-1}b\) of a fibration \(p : E \to B\).