Homology, Homotopy and Applications
Volume 18 (2016)
Dwyer–Kan localization revisited
Pages: 27 – 48
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