A characterization of simplicial localization functors and a discussion of DK equivalences

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On the Unicity of the Homotopy Theory of Higher Categories (2011, methodology): “j−1×Ck−1)) (f(C i+k ∪ Ci C i+j)) →f(Ci+j ×Ci Ci+k), is an equivalence. 18 CLARK BARWICK AND CHRISTOPHER SCHOMMER-PRIES (d) For any 0 ≤k≤n, the natural morphism (f(Ck) ⊔f(Ck)) ∪(f(σ kC 1)⊔f(σkC1)) f(σk[3]) →f(C k) in Cis an equivalence. (C.4) Universality. For any quasicategory D and any fully faithf”
Presenting higher stacks as simplicial schemes (2013, methodology): “ypergroupoid π: X˜ →X, the functor Lπ∗: cM(X)cart →cM(X˜)cart 30 J.P.PRIDHAM induces weak equivalences of classifying diagrams. This makes it a weak equivalence of relative categories in the sense of [BK1]. Deﬁnition 5.11. For any object SpecA ∈AﬀC, there is a model category Mod(A) of A-modules in C”
On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “It shows that the functor ν induces an equivalence of relative categories, in the language of [3], or equivalently, an equivalence between the simplicial localisations of the categories involved [15, 2].”
Pro-categories in homotopy theory (2015, result): “Rezk(C,W) (in the complete Segal space model structure). The fact that this construction is equivalent to the two constructions described above can be proven, for example, by combining the results of [BK12a] and [BK12b] with To¨en’s unicity theorem [To05, Th´eor`em 6.3]. For our purposes, we will on”