Oplax actions and enriched icons with applications to coalgebroids and quantum categories

This thesis is a formal treatise around the concept of bialgebroids and the alternative ways to describe them. There are two characterizations of bialgebroids due to Szlachányi which play a central role in our investigation; one as cocontinuous opmonoidal monads on the category of two-sided R-modules, and another as certain skew monoidal structures on the category of right R-modules. Lack and Street internalised Szlachányi's characterization to a suitable monoidal bicategoryM. In this way, they obtain an equivalence between opmonoidal monads on the enveloping monoidale induced by a biduality and right skew monoidales whose unit has a right adjoint in M. Such equivalence provides a characterization of the quantum categories defined by Day and Street. In the first two chapters, we focus on the simpler structure of a coalgebroid. In a monoidal bicategory M, coalgebroids generalise as opmonoidal arrows between enveloping monoidales. Chapter 2 has two main results. The first one, following Lack and Street's methods, is a characterisation of opmonoidal arrows between enveloping monoidales which leads to the new concept of oplax actions with respect to a skew monoidale. The second result involves the study of comodules. Comodules for coalgebroids are classically defined as comodules for the underlying coring; and in a monoidal bicategory M, opmonoidal arrows and oplax actions each admit a notion of comodule. We prove that these three ways to define comodules are equivalent. The equivalence between opmonoidal arrows and oplax actions mentioned above is analogous to that of opmonoidal monads and right skew monoidales. We formalise this statement in Chapter 3, and show along the way that monads of oplax actions are right skew monoidales whose unit has a right adjoint. The last chapter focuses on a different characterisation of bialgebroids: Moerdijk proved that a monad on a monoidal category is an opmonoidal monad if and only if the category of algebras has a monoidal structure such that the forgetful functor is strong monoidal. In other words, the 2-category OpMon of monoidal categories, opmonoidal functors, and opmonoidal natural transformations has Eilenberg-Moore objects for monads. We generalise this theorem in two directions: a multi-object version, and a version enriched in a monoidal bicategory. For the multi-object version, we replace OpMon with the 2-category Icon of bicategories, oplax functors, and icons. And for the version enriched in a monoidal bicategory M, we replace Icon with a bicategory Icon(M) of M-enriched bicategories, M-enriched oplax functors, and M-enriched icons. At this level of generality, the theorem asserts that the bicategory Icon(M) has Eilenberg-Moore objects for monads if M does.