The hammock localization preserves homotopies

TL;DR

This paper demonstrates that the hammock localization preserves homotopies and explores its applications in homotopy theory, including properties of homotopy idempotent functors and Bousfield localizations, under Vopěnka's principle.

Contribution

It proves that hammock localization preserves homotopies and applies this to characterize homotopy idempotent functors and establish the existence of Bousfield localizations under certain set-theoretic assumptions.

Findings

Homotopies induce homotopies after hammock localization.

Homotopy idempotent functors are characterized by simplicial orthogonality under Vopěnka's principle.

Left Bousfield localizations exist in certain model categories assuming Vopěnka's principle.

Abstract

The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We describe applications of this fact to the study of homotopy algebras over monads and homotopy idempotent functors. Among other things, we prove that, under Vop\v{e}nka's principle, every homotopy idempotent functor in a cofibrantly generated model category is determined by simplicial orthogonality with respect to a set of morphisms. We also give a new proof of the fact that left Bousfield localizations with respect to a class of morphisms always exist in any left proper combinatorial model category under Vop\v{e}nka's principle.