Pre-torsors and Galois comodules over mixed distributive laws

TL;DR

This paper develops a generalized bi-Galois theory for comonads arising from mixed distributive laws, establishing new categorical equivalences and extending classical results to a broader algebraic context.

Contribution

It introduces a novel equivalence between pre-torsors over regular adjunctions and regular comonad arrows, generalizing known Galois object relationships.

Findings

Establishes an equivalence between categories of pre-torsors and regular comonad arrows.

Develops a bi-Galois theory connecting comodule categories of different comonads.

Generalizes classical Galois theory for coalgebras to a categorical framework.

Abstract

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions $(N_A,R_A)$ and $(N_B,R_B)$ on one hand, and the category of regular comonad arrows $(R_A,\xi)$ from some equalizer preserving comonad ${\mathbb C}$ to $N_BR_B$ on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras.Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad ${\mathbb D}$ and a co-regular comonad arrow from ${\mathbb D}$ to $N_A R_A$, such that the comodule categories of ${\mathbb C}$ and ${\mathbb D}$ are equivalent.