Bicategory of entwinings

TL;DR

This paper constructs a bicategory framework for entwinings over variable rings, establishing a canonical morphism to the bicategory of corings, thus advancing the categorical understanding of algebraic structures.

Contribution

It introduces a bicategory of entwinings with detailed morphisms and 2-cells, and links it canonically to the bicategory of corings, enriching the categorical theory of algebraic structures.

Findings

Defined a bicategory of entwinings over variable rings.

Established a canonical morphism to the bicategory of corings.

Provided a categorical framework connecting entwinings and corings.

Abstract

We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the "composed coring" from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Street's bicategory of corings.