Equipping weak equivalences with algebraic structure

TL;DR

This paper explores how weak equivalences in model categories can be endowed with algebraic structures, establishing conditions and constructions that characterize such structures for various types of equivalences.

Contribution

It introduces a general framework using algebraic injectivity and cone injectivity to equip weak equivalences with algebraic structures in model categories.

Findings

Existence of a monad characterizing weak homotopy equivalences as T-algebras.

Extension of the framework to quasi-isomorphisms and other equivalences.

General results on algebraic structures for weak equivalences in combinatorial model categories.

Abstract

We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.