Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

Dwyer–Kan localization revisited

Pages: 27 – 48

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


Vladimir Hinich (Department of Mathematics, University of Haifa, Mount Carmel, Haifa, Israel)


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.


DK localization, infinity-category

2010 Mathematics Subject Classification

18D20, 55U35

Published 31 May 2016