Coalgebroids in monoidal bicategories and their comodules

Abstract

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 $R⊣R^{\circ }$, or enveloping monoidale, in a monoidal bicategory of modules $\mathsf{Mod}(\mathcal{V})$ for a monoidal category $\mathcal{V}$. Lack and Street proved that quantum categories are in equivalence with right skew monoidales whose unit has a right adjoint in $\mathsf{Mod}(\mathcal{V})$. Our first important result is similar to that of Lack and Street. It is a characterisation of opmonoidal \emph{arrows} on enveloping monoidales in terms of a new structure named \emph{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.

63 pages. v2: theorem numbering changed and minor corrections. Final journal version

References in corpus (2)

• Skew-monoidal categories and bialgebroids
• Morita contexts as lax functors

Cited by in corpus (1)

• Oplax Hopf Algebras