Revisiting function complexes and simplicial localisation

TL;DR

This paper clarifies the nature of homotopy function complexes, showing they can be derived as functors, compute hom-spaces in simplicial localisations, and be calculated via fibrant replacements in localized model structures.

Contribution

It establishes new theoretical connections between homotopy function complexes, derived functors, and fibrant replacements in model categories.

Findings

Homotopy function complexes are total right derived functors.

They compute hom-spaces in simplicial localisation.

Can be calculated using fibrant replacements in localized model structures.

Abstract

In this paper three results are established: firstly, that the homotopy function complexes of Dwyer and Kan can be defined as certain total right derived functors; secondly, that they functorially compute the homotopy type of the hom-spaces in the simplicial localisation; and thirdly, that they can be computed by fibrant replacements in a suitable left Bousfield localisation of the projective model structure on simplicial presheaves.