Simple transitive 2-representations via (co)algebra 1-morphisms

TL;DR

This paper develops a method to construct simple transitive 2-representations of fiat 2-categories using algebra and coalgebra 1-morphisms, extending Morita-Takeuchi theory with explicit examples.

Contribution

It introduces a novel approach to constructing simple transitive 2-representations via (co)algebra 1-morphisms and extends Morita-Takeuchi theory to this framework.

Findings

Construction of simple transitive 2-representations using coalgebra 1-morphisms.

Dual construction using algebra 1-morphisms.

Explicit examples illustrating the extended Morita-Takeuchi theory.

Abstract

For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples explicitly.