Left Kan extensions that are algebraic over colax-idempotent 2-monads

TL;DR

This paper generalizes classical results on finite-product-preserving left Kan extensions to those that preserve algebraic structures defined by suitable colax-idempotent 2-monads, using double category language.

Contribution

It extends the theory of Kan extensions to algebraic structures governed by colax-idempotent 2-monads within the framework of double categories.

Findings

Generalization of classical results to colax-idempotent 2-monads

Introduction of double category approach for Kan extensions

Analysis of induced algebra structures on presheaf objects

Abstract

Using the language of double categories we generalise a classical result on finite-product-preserving left Kan extensions, by Ad\'amek and Rosick\'y, to one on left Kan extensions that preserve algebraic structures defined by `suitable' colax-idempotent 2-monads, as well as obtain two related results. To be precise, by `suitable' 2-monads here we mean ones that extend to normal lax double monads. In an appendix we consider induced algebra structures on presheaf objects.