The hammock localization preserves homotopies

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 also show that, under Vopěnka’s principle, every homotopy idempotent functor in a cofibrantly generated model category is determined by simplicial orthogonality with respect to a set of morphisms. Finally, we 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 Vopěnka’s principle.

Keywords

model category, homotopy function complex, localization, homotopy algebra

2010 Mathematics Subject Classification

18C35, 55P60, 55U35

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Published 3 December 2015