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 ,ξ) 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.
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Böhm, G., Menini, C. Pre-torsors and Galois Comodules Over Mixed Distributive Laws. Appl Categor Struct 19, 597–632 (2011). https://doi.org/10.1007/s10485-008-9185-9
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DOI: https://doi.org/10.1007/s10485-008-9185-9