Idempotent splittings, colimit completion, and weak aspects of the theory of monads

TL;DR

This paper explores how recent 'weak' generalizations in monad theory can be systematically understood through the lens of enriched category theory, specifically using categories enriched over Cauchy complete categories.

Contribution

It introduces a systematic approach to weak monad generalizations by leveraging categories enriched in Cauchy complete categories, bridging recent constructions with enriched category theory.

Findings

Weak generalizations are formalized via enrichment in Cauchy complete categories.

A systematic framework for weak monad constructions is developed.

Connections between recent literature and enriched category theory are clarified.

Abstract

We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.