Pre-torsors and Galois comodules over mixed distributive laws

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 $N_B{R}_B$ 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.

34 pages LaTeX file. v2: a few typos corrected

References in corpus (1)

• Coalgebra Gauge Theory

Cited by in corpus (1)

• Morita contexts as lax functors