Diagrams indexed by Grothendieck constructions

Hollander, Sharon

doi:10.4310/hha.2008.v10.n3.a10

Cited by 4 publications

(4 citation statements)

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“…The Grothendieck construction has appeared elsewhere in homotopy theory, suggesting many possible applications of our work. When B is an indexing category, composing Φ with the nerve functor N : Cat → sS et, from the category of small categories to the category of simplicial sets, gives rise, via the Grothendieck construction, to a model for the homotopy colimit of the diagram Φ as the geometric realization of the simplicial replacement of Φ [BK72,Hol08]. It provides a way to shift from the study of colored operads with a fixed set of colors to the study of colored operads in general and an analogous shift in the setting of enriched categories [Sta12].…”

Section: Introductionmentioning

confidence: 99%

Algebras of Higher Operads as Enriched Categories

Batanin

Weber

2008

Appl Categor Struct

101

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One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39-103, 1998) to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.

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Section: Introductionmentioning

confidence: 99%

Algebras of Higher Operads as Enriched Categories

Batanin

Weber

2008

Appl Categor Struct

101

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“…This is a well-known fact; for example, in case A is a group G, this reproduces the equivalence of [3] between the model categories of simplicial sets with G-action and simplicial sets over the classifying space BG. See also [7,9]. Corollary E. If A is a groupoid, the functor h !…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Left fibrations and homotopy colimits

Heuts

Moerdijk

2014

Math. Z.

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Abstract. For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with the covariant model structures. Also, we show that versions of Quillen's Theorems A and B for ∞-categories easily follow.

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“…C W S C op ! .S#NC/ is not a right Quillen equivalence, between S C op and the closed model category of pointed objects over NC (ie, of retractive simplicial sets over NC ), except for example, when C is a groupoid (see Hollander [38,Theorem 2.7], or Dror, Dwyer and Kan [23] for C D G a group). The closed model structure on .S#NC/ is induced by the usual one of simplicial sets; that is, where a map is a weak equivalence, cofibration or fibration if and only if it is a weak equivalence, cofibration or fibration of simplicial sets, respectively.…”

Section: Cohomology Of Diagrams Of Simplicial Setsmentioning

confidence: 99%