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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