Coalgebroids in monoidal bicategories and their comodules

Ramon Abud Alcala

Quantum categories have been recently studied because of their relation to bialgebroids, small categories, and skew monoidales. This is the first of a series of papers based on the author's PhD thesis in which we examine the theory of quantum categories developed by Day, Lack, and Street. A quantum category is an opmonoidal monad on the monoidale associated to a biduality $R\dashv R^{o}$, or enveloping monoidale, in a monoidal bicategory of modules $\Mod(V})$ for a monoidal category $V$. Lack and Street proved that quantum categories are in equivalence with right skew monoidales whose unit has a right adjoint in $\Mod(V)$. Our first important result is similar to that of Lack and Street. It is a characterisation of opmonoidal arrows on enveloping monoidales in terms of a new structure named oplax action. We then provide three different notions of comodule for an opmonoidal arrow, and using a similar technique we prove that they are equivalent. Finally, when the opmonoidal arrow is an opmonoidal monad, we are able to provide the category of comodules for a quantum category with a monoidal structure such that the forgetful functor is monoidal.

Keywords: coalgebroid, comodules, monoidale, skew monoidale, oplax action, monoidal bicategory, bicategory, quantum category, bialgebroid

2010 MSC: 18D05, 18D10, 16T15}

Theory and Applications of Categories, Vol. 33, 2018, No. 30, pp 898-963.

Published 2018-08-30.

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