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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### Dwyer–Kan localization revisited

Pages: 27 – 48

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a3

#### Author

#### Abstract

A version of Dwyer–Kan localization in the context of $\infty$-categories and simplicial categories is presented. Some results of the classical papers—“Simplicial localizations of categories” [*J. Pure Appl. Algebra* 17 (1980), no. 3, 267–284], “Calculating simplicial localizations” [*J. Pure Appl. Algebra* 18 (1980), no. 1, 17–35], and “Function complexes in homotopical algebra” [*Topology* 19 (1980), no. 4, 427–440]—are reproven and generalized. We prove that a Quillen pair of model categories gives rise to an adjoint pair of their DK localizations (considered as $\infty$-categories). We study families of $\infty$-categories and present a result on localization of a family of $\infty$-categories. This is applied to localization of symmetric monoidal $\infty$-categories where we were able to get only partial results.

#### Keywords

DK localization, infinity-category

#### 2010 Mathematics Subject Classification

18D20, 55U35