Comma 2-comonad I: Eilenberg-Moore 2-category of colax coalgebras

TL;DR

This paper develops a detailed framework for the Eilenberg-Moore 2-category of colax coalgebras using a comma 2-comonad, integrating key categorical constructions into this setting.

Contribution

It introduces a comma 2-comonad on a 2-category of functors and fully describes its Eilenberg-Moore 2-category of colax coalgebras, connecting many core concepts in formal category theory.

Findings

Complete description of the Eilenberg-Moore 2-category of colax coalgebras

Integration of adjoint triples, distributive laws, and Frobenius functors into the framework

Unified categorical setting for fundamental constructions

Abstract

In this paper we describe a comma 2-comonad on the 2-category whose objects are functors, 1-cell are colax squares and 2-cells are their transformations. We give a complete description of the Eilenberg-Moore 2-category of colax coalgebras, colax morphisms between them and their transformations and we show how many fundamental constructions in formal category theory like adjoint triples, distributive laws, comprehension structures, Frobenius functors etc. naturally fit in this context.