A characterization of simplicial localization functors and a discussion of DK equivalences
The paper under review belongs to the study of the ``homotopy theory of homotopy theories'', a survey of which can be found in [\textit{J. E. Bergner}, IMA Vol. Math. Appl. 152, 69--83 (2010; Zbl 1200.18011)]. There are several models to describe a homotopy of homotopy theories, including the following: - Relative categories \({\mathcal R}el{\mathcal C}at\), i.e., categories with specified weak equivalences; - Simplicial categories \({\mathcal S}{\mathcal C}at\), i.e., categories enriched in simplicial sets; - Complete Segal spaces, which are the fibrant objects in the Rezk model structure on simplicial spaces. There are constructions going between the different models. For example, starting with a relative category \((C,W)\), the Dwyer-Kan hammock localization \(L^H : {\mathcal R}el{\mathcal C}at \to {\mathcal S}{\mathcal C}at\) produces a simplicial category that encodes the same homotopy theory as \((C,W)\). In a previous paper [\textit{C. Barwick} and \textit{D. M. Kan}, Indag. Math., New Ser. 23, No. 1--2, 42--68 (2012; Zbl 1245.18006)], the authors lifted the Rezk model structure on simplicial spaces to a Quillen equivalent one on the category \({\mathcal R}el{\mathcal C}at\) of relative categories. Weak equivalences in both cases are thus called Rezk equivalences. However, the notion of Rezk equivalence between relative categories remained somewhat obscure. The main result of the current paper is to fix that problem by providing a more concrete description. Theorem 1.8 states that a map of relative categories is a Rezk equivalence if and only if it is a DK-equivalence, which means that it becomes a DK-equivalence in \({\mathcal S}{\mathcal C}at\) upon applying the hammock localization. Here, a map of simplicial categories is a DK-equivalence if it induces a weak equivalence on all mapping spaces and an equivalence of homotopy categories. The other main result deals with the arbitrariness of having used the hammock localization in the definition above. There are other constructions \(L : {\mathcal R}el{\mathcal C}at \to {\mathcal S}{\mathcal C}at\) which are naturally DK-equivalent to \(L^H\). Such constructions, called \textit{simplicial localizations}, are just as good as \(L^H\) and can be used in its stead. The authors provide a characterization of such functors \(L : {\mathcal R}el{\mathcal C}at \to {\mathcal S}{\mathcal C}at\). Theorem 1.7 states that a functor \(L : {\mathcal R}el{\mathcal C}at \to {\mathcal S}{\mathcal C}at\) sending DK-equivalences to DK-equivalences is a simplicial localization if and only if it is a homotopy inverse to the Dwyer-Kan delocalization (or relativization) functor \({\mathcal R}el{\mathcal C}at\).
scientific article; zbMATH DE number 6009287
scientific article