Model structures from a monad on presheaves

TL;DR

This paper identifies conditions under which monad algebras on presheaf categories form fibrant objects in a model structure, establishing a minimal model structure where monad units serve as fibrant replacements.

Contribution

It provides a framework linking monad algebras on presheaves to fibrant objects in model categories, defining minimal conditions for such structures.

Findings

Conditions for monad algebras to be fibrant in a model structure

The resulting model structure is minimal with monad units as fibrant replacements

Establishment of a connection between monad structures and model category fibrancy

Abstract

In this note we describe conditions under which the algebras for a monad on a presheaf category equipped with some additional structure are fibrant objects in a model structure. We also prove that when these conditions are satisfied the resulting model structure is, in a suitable sense, the smallest model structure for which the units of the monad give a fibrant replacement.