Homotopy-theoretic aspects of 2-monads

TL;DR

This paper explores the homotopical properties of 2-monads and their algebras using enriched model categories, establishing canonical and new model structures on 2-categories and 2-monads.

Contribution

It introduces a Cat-enriched model category framework for 2-monads, connecting homotopy theory with 2-categorical algebra and constructing new model structures.

Findings

Canonical model structure on 2-categories with finite limits and colimits

Model structures on 2-category of algebras for a 2-monad

Model structure on 2-category of 2-monads

Abstract

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K.