Simplicial Homotopy Theory

Goerss, Paul G.; Jardine, John F.

doi:10.1007/978-3-0346-0189-4

Cited by 408 publications

(235 citation statements)

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“…The functor Ex ∞ : SSet → SSet is the finitely accessible fibrant replacement simplicial functor (see [17]). For a small simplicial functor F : A op → SSet, let Fib(F ) be the composition…”

Section: Class-combinatorial Model Categoriesmentioning

confidence: 99%

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Generalized parton distributions of the photon for nonzero skewness

Mukherjee

Nair

2012

Physics Letters B

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Abstract. We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories.The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [10], where we introduced the concepts of locally class-presentable and class-accessible categories.In this work we prove that the category of weak equivalences of a nice class-combinatorial model category is class-accessible. Our extension of J. Smith localization theorem depends on the verification of a cosolution-set condition. The deepest result is that the (left Bousfield) localization of a class-combinatorial model category with respect to a strongly class-accessible localization functor is class-combinatorial again.

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Section: Class-combinatorial Model Categoriesmentioning

confidence: 99%

“…On the other hand, every F -local equivalence between F -local objects is a weak equivalence in K (cf. [17] X.2.1. 2)).…”

Section: Left Bousfield Localizationsmentioning

confidence: 99%

Generalized parton distributions of the photon for nonzero skewness

Mukherjee

Nair

2012

Physics Letters B

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“…We will use freely the language of homotopical algebra. The reader unfamiliar with this language is invited to consult Hirschhorn [20], Hovey [21], Goerss and Jardine [15] or Quillen's original monograph [27]. ].…”

Section: Preliminariesmentioning

confidence: 99%

Generalized spectral categories, topological Hochschild homology and trace maps

Tabuada¹

2010

Algebr. Geom. Topol.

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Given a monoidal model category C and an object K in C , Hovey constructed in [22] the monoidal model category Sp † .C; K/ of K -symmetric spectra over C . In this paper we describe how to lift a model structure on the category of C -enriched categories to the category of Sp † .C; K/-enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to H Z-categories.As an application we obtain: (1) the invariance under weak equivalences of the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of dg categories; (2) non-trivial natural transformations from algebraic K -theory to THH . 55P42, 18D20, 18G55; 19D55

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“…It it shown in Lemma II.8.8 in [GJ99,p. 127] that a category of cofibrant objects satisfies the gluing lemma.…”

Section: The Gluing Lemmamentioning

confidence: 85%

“…The notion of a cylinder object in [GJ99, p. 123] is slightly different from our notion, but as the cylinder axiom 6.4.2 holds our cylinder functor from Section 6.2 applied to the identity yields a cylinder object in the sense of [GJ99,p. 123].…”

Section: Pushout Diagrams Of the Formmentioning

confidence: 99%

Controlled algebra for simplicial rings and algebraic K-theory

Ullmann¹

Geometric and Cohomological Group Theory

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We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K-theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is the first step to prove the algebraic K-theory isomorphism conjecture for simplicial rings. We construct a category of controlled simplicial modules, show that it has the structure of a Waldhausen category and discuss its algebraic K-theory.We lay emphasis on detailed proofs. Highlights include the discussion of a simplicial cylinder functor, the gluing lemma, a simplicial mapping telescope to split coherent homotopy idempotents, and a direct proof that a weak equivalence of simplicial rings induces an equivalence on their algebraic K-theory. Because we need a certain cofinality theorem for algebraic K-theory, we provide a proof and show that a certain assumption, sometimes omitted in the literature, is necessary. Last, we remark how our setup relates to ring spectra.

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Simplicial Homotopy Theory

Cited by 408 publications

References 0 publications

Generalized parton distributions of the photon for nonzero skewness

Generalized parton distributions of the photon for nonzero skewness

Generalized spectral categories, topological Hochschild homology and trace maps

Controlled algebra for simplicial rings and algebraic K-theory

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