THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES

By Julia E. Bergner. 2005

156 citations recorded by the source corpus.

Abstract Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.

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

View Paper: https://www.semanticscholar.org/paper/c28719b46fdf29310c8d5fdb9345eb541a81e3ed

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Citations in the corpus

Dendroidal sets as models for homotopy operads (2009, methodology): “We denote by J the nerve of the contractible groupoid with two objects 0 and 1 (that is, J is the nerve of the fundamental groupoid of Δ[1]).”
Moduli Problems for Ring Spectra (2011, methodology): “If X is any topological space, then the n-category π≤nX is an example of an n-groupoid: for example, the 1-morphisms in π≤nX are given by paths p : [0, 1] → X, and every path p has an inverse q (up to homotopy) given by q(t) = p(1 − t).”
Dendroidal Segal spaces and ∞‐operads (2013, unknown): “Moreover, they are all Quillen equivalent to the model category of simplicial categories discovered by Bergner [2], thus providing a strictification or rigidification result for each of these notions of category-up-to-homotopy.”
Rigidification of algebras over multi-sorted theories (2005, methodology): “In [ 4 ], we use the ideas of this proof to prove an analogous relationship between the category of all small simplicial categories and the category of all Segal categories.”
A model category structure on the category of simplicial categories (2004, methodology): “In [1], we show that this model structure is Quillen equivalent to three other model category structures.”
Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, methodology): “Bergner considers further this theory and shows the equivalence with simplicial categories and Segal categories [34].”
Presenting higher stacks as simplicial schemes (2013, background): “ial sets for all i. Remark 5.13. In [TV2], N(Mod)0 is denoted by QCoh, and RHomAﬀ∼,τ C (|X|,N(Mod)0) is regarded as nerve of the ∞-groupoid of quasi-coherent sheaves on |X|. Using the equivalences of [Ber1], we should thus regard the complete Segal space i → RHomAﬀ∼,τ C (|X|,N(Mod)i) as the ∞-catego”
∞-Categories for the Working Mathematician (2018, methodology): “But if in an outer horn Λ0[2] → A or Λ2[2] → A, the initial or final edges, respectively, are isomorphisms, then intuitively a filler should exist a1 a1 a0 a2 a0 a2 hf−1 g ∼ f ∼”
(Infinity,2)-Categories and the Goodwillie Calculus I (2009, background): “Bergner has shown that this functor is a right Quillen equivalence (see [10]; we will prove a generalization of this result as Proposition 2.”
Higher Hochschild Homology, Topological Chiral Homology and Factorization Algebras (2010, background): “Indeed simplicial categories are model for (∞, 1)-categories [5].”
Dendroidal sets and simplicial operads (2011, background): “The category PreCat of (Segal) precategories studied by Bergner in [3] is simply PreOper/η.”