Skew-monoidal categories and bialgebroids

TL;DR

This paper establishes a precise correspondence between skew-monoidal structures on right R-modules and right bialgebroids, extending the theory to bimodules and monoidal lax comonads for a broader understanding of bialgebroid comodules.

Contribution

It characterizes closed skew-monoidal structures on right R-modules as right bialgebroids and explores their induced structures and relations with opmonoidal monads and monoidal lax comonads.

Findings

Skew-monoidal structures on R-modules are exactly right bialgebroids.

Induced quotient skew-monoidal structures on R-R-bimodules.

Connection between opmonoidal monads and skew-monoidal structures with a natural distributive law.

Abstract

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R-R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.