Pre-torsors and Galois Comodules Over Mixed Distributive Laws |
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| Authors: | Gabriella Böhm Claudia Menini |
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| Affiliation: | 1.Research Institute for Particle and Nuclear Physics,Budapest,Hungary;2.Department of Mathematics,University of Ferrara,Ferrara,Italy |
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| 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{\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{\mathbb D} and a co-regular comonad arrow from
\mathbb D{\mathbb D} to N
A
R
A
, such that the comodule categories of
\mathbb C{\mathbb C} and
\mathbb D{\mathbb D} are equivalent. |
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| Keywords: | |
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