A Characterization of Simplicial Localization Functors and a Discussion of Dk Equivalences

In a previous paper we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of “relative categories.” Here, we characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author. We employ this characterization to obtain a more explicit description of the weak equivalences in the model category of relative categories mentioned above by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations. 1. An overview We start with some preliminaries. 1.1. Relative categories. As in [BK] we denote by RelCat the category of (small) relative categories and relative functors between them, where by a relative category we mean a pair (C,W ) consisting of a category C and a subcategory W ⊂ C which contains all the objects of C and their identity maps and of which the maps will be referred to as weak equivalences and where by a relative functor between two such relative categories we mean a weak equivalence preserving functor. 1.2. Rezk equivalences. In [BK] we lifted Charles Rezk’s complete Segal model structure on the category sS of (small) simplicial spaces (i.e. bisimplicial sets) to a Quillen equivalent model structure on the category RelCat (1.1). We will refer to the weak equivalences in both these model structures as Rezk equivalences and denote by both Rk ⊂ sS and Rk ⊂ RelCat the subcategories consisting of these Rezk equivalences. 1.3. Homotopy equivalences between relative categories. A relative functor f : X → Y between two relative categories (1.1) is called a homotopy equivalence if there exists a relative functor g : Y → X (called a homotopy inverse of f) such that the compositions gf and fg are naturally weakly equivalent (i.e. can be connected by a finite zigzag of natural weak equivalences) to the identity functors of X and Y respectively. Date: July 28, 2011. 1 2 C. BARWICK AND D. M. KAN 1.4. DK-equivalences. A map in the category SCat of simplicial categories (i.e. categories enriched over simplicial sets) is [Be1] called a DK-equivalence if it induces weak equivalences between the simplicial sets involved and an equivalence of categories between their homotopy categories, i.e. the categories obtained from them by replacing each simplicial set by the set of its components. Furthermore a map in RelCat will similarly be called a DK-equivalence if its image in SCat is so under the hammock localization functor [DK2] L : RelCat −→ SCat (or of course the naturally DK-equivalent functors RelCat→ SCat considered in [DK1] and [DHKS, 35.6]). We will denote by both DK ⊂ SCat and DK ⊂ RelCat the subcategories consisting of these DK-equivalences. Next we define what we mean by 1.5. Simplicial localization functors. In defining DK-equivalences in RelCat (1.4) we used the hammock localization functor and not one of the other DKequivalent functors mentioned because, for our purposes here it seemed to be the more convenient one. However in other situations the others are more convenient and it therefore makes sense to define in general a simplicial localization functor as any functor RelCat → SCat which is naturally DK-equivalent to the functors mentioned above (1.4).