Pre-torsors and Galois Comodules Over Mixed Distributive Laws

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 (NA, RA) and (NB, RB) on one hand, and the category of regular comonad arrows (RA, ξ) from some equalizer preserving comonad C to NBRB 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 D and a co-regular comonad arrow from D to NARA, such that the comodule categories of C and D are equivalent.