Fibrations of algebras

TL;DR

This paper investigates fibrations derived from indexed categories of algebras, unifying various mathematical examples through a semidirect product perspective and establishing a canonical action of the base category on fibers over an initial object.

Contribution

It introduces a general framework for fibrations of algebras from functors, unifies diverse examples, and proves a canonical action of the base category on fibers over an initial object.

Findings

Unified various algebraic fibrations under a semidirect product perspective.

Established a canonical action of the base category on fibers over an initial object.

Connected the results to classical actions like the fundamental group on fibers.

Abstract

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 $p : E \to B$.