Relative categories: Another model for the homotopy theory of homotopy theories

By Clark Barwick, Donghui Kan. 2010

169 citations recorded by the source corpus.

We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/44aa5760ea54ce866d1c3de63af2f871c808f5c6

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Citations in the corpus

Lax colimits and free fibrations in ∞-categories (2017, methodology): “This follows from [BK12, 7.5], since any ﬁbrant replacement functor gives a homotopy equivalence of relative categories.”
A note on the (∞,n)–category of cobordisms (2015, methodology): “Following [BK12] a rather weak notion of p8, 1q-category is given by relative categories.”
Shifted Poisson and symplectic structures on derived N ‐stacks (2015, methodology): “Our model of ∞-categories will be given by the relative categories of [BK] — these consist of a category C and a subcategory W (whose morphisms we will refer to as weak equivalences).”
On the algebraic K‐theory of higher categories (2012, background): “ll often write C for the simplicial nerve N(C∆) [43, Df. 1.1.5.5], which is an ∞-category [43, Pr. 1.1.5.10]. It will also be convenient to have a model of various ∞-categories as relative categories [4]. To make this precise, we recall the following. 1.5. Deﬁnition. A relative category is an ordina”
Sheaves of categories and the notion of 1-affineness (2013, background): “low are the notation and conventions we use in this paper. They can easily be skipped. • our models for (∞,1)-categories are categories with weak equivalences (a-k-a relative categories). We refer to [4] for the details about the homotopy theory of relative categories. • we write hC for the homotopy”
Lax Colimits and Free Fibrations in $\infty$-Categories (2015, background): “ets N(2,1)C → N(2,1)D can be identiﬁed with a normal pseudofunctor C → D. Deﬁnition A.22. Recall that a relative category is a category C equipped with a subcategory Wcontaining all isomorphisms; see [BK12] for a more extensive discussion. A functor of relative categories f: (C,W) → (C′,W′) is a fun”
Pro-categories in homotopy theory (2015, result): “n 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 only need the ”
From operator categories to higher operads (2013, background): “≃ Map α X⊗(x,y) for any objects x∈ X⊗ I , y∈ X ⊗ {1}, and any collection {x xi | i∈ |I|} of inert morphisms. In other words, C, when restricted to Φ-quasioperads is a relative functor in the sense of [4], and it reﬂects weak equivalences. In fact, it is part of a Quillen pair. 10.14.1. Corollary. Su”
A universal characterization of higher algebraic K-theory (2010, methodology): “Let M : CatS → CatS denote the composite functor (3.11) M : CatS ψtri // Catex∆ Spec // CatS The work above shows that M induces a simplicial functor on the Dwyer-Kan simplicial localizations and therefore descends to the level of ∞-categories to produce a functor NM : NLH(CatS) −→ NL H(CatS).”
Higher quasi-categories vs higher Rezk spaces (2012, result): “Another model, relative categories, was introduced and compared to the other models by Barwick and Kan in [6].”