[Paper Review] Coalgebroids in monoidal bicategories and their comodules

Ramón Abud Alcalá|arXiv (Cornell University)|Feb 6, 2018

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper establishes a unified framework for comodules over opmonoidal arrows in monoidal bicategories, introducing oplax actions as a key tool. It proves three notions of comodule are equivalent and shows that when the opmonoidal arrow is a monad, the category of comodules inherits a monoidal structure via a strong monoidal forgetful functor—extending quantum category theory to a broader bicategorical setting.

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 $R\dashv R^{\circ}$, 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.

Motivation & Objective

• To generalize the theory of bialgebroids and quantum categories beyond classical module-theoretic settings by using 2-categorical tools.
• To provide a unified treatment of comodules for opmonoidal arrows in monoidal bicategories, particularly in the context of enveloping monoidales.
• To establish that three distinct definitions of comodules for opmonoidal arrows are equivalent, thereby clarifying foundational structures.
• To show that when the opmonoidal arrow is an opmonoidal monad, the category of comodules becomes monoidal with a strong monoidal forgetful functor.
• To extend the results of Day, Lack, and Street on quantum categories to a more general bicategorical framework using opmonadic adjunctions and oplax actions.

Proposed method

• Introduces oplax actions as a new structure to characterize opmonoidal arrows on enveloping monoidales, generalizing the notion of opmonoidal monads.
• Uses the equivalence between opmonoidal arrows and oplax actions to define and compare three distinct notions of comodules for such arrows.
• Applies the theory of opmonadic adjunctions and dual adjunctions to relate comodules to modules over associated monads and comonads.
• Employs the Kleisli construction and comparison functors to prove fullness and essential surjectivity, establishing equivalences between categories of comodules.
• Leverages the monoidal bicategory structure of Mod(V) and the biduality R ⊣ R◦ to define enveloping monoidales R◦⊗R and lift monoidal structures to comodule categories.
• Uses horizontal composition and coherence isomorphisms in the bicategory to construct the associator and unitor in the monoidal structure on comodules.

Experimental results

Research questions

• RQ1 How can comodules for opmonoidal arrows in monoidal bicategories be defined in multiple equivalent ways?
• RQ2 What is the role of oplax actions in characterizing opmonoidal arrows on enveloping monoidales?
• RQ3 Under what conditions does the category of comodules for an opmonoidal monad inherit a monoidal structure?
• RQ4 How do the standard definitions of comodules for coalgebroids and bialgebroids relate to the new framework via opmonoidal arrows?
• RQ5 Can the forgetful functor from the category of comodules to the base category be monoidal, and under what conditions?

Key findings

• Three distinct definitions of comodules for an opmonoidal arrow are proven to be equivalent via a sequence of isomorphisms and equivalences in the bicategory.
• The category of comodules for an opmonoidal monad on an enveloping monoidale inherits a monoidal structure such that the forgetful functor to the base category is strong monoidal.
• The monoidal structure on comodules is constructed using horizontal composition and coherence isomorphisms, with the associator and unitors induced from the monad's structure.
• For the motivating case M = Mod_k, the results recover and unify known definitions of comodules for coalgebroids and bialgebroids, showing equivalence across oplax actions, standard coactions, and opmonoidal arrows.
• The equivalence between comodules defined via oplax actions and those defined via coactions is established, with the only difference being the underlying module category (R-Mod-S vs. Mod-S).
• Corollary 7.19 confirms that for an opmonoidal monad B on an enveloping monoidale R◦R, the category rComodB((R, e1), (R, e1)) is monoidal with a strong monoidal forgetful functor to M(R, R).