On the equivalence between Lurie's model and the dendroidal model for infinity-operads

Heuts, Gijs; Hinich, Vladimir; Moerdijk, Ieke

doi:10.1016/j.aim.2016.07.021

Cited by 38 publications

(65 citation statements)

References 32 publications

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“…The same result holds when only one of the two arrows is a cofibration. Indeed, this follows from the fact that the operadic model structure is left proper [7], the covariant model structures over weakly equivalent dendroidal sets are Quillen equivalent [11] and the fact that two (naturally) Quillen equivalent diagrams of model categories have Quillen equivalent homotopy pullbacks [2].…”

Section: Applicationsmentioning

confidence: 99%

“…One may therefore expect a similar gluing result to hold for fibrations between dendroidal sets which have the same locality property. Operadic fibrations do not have this property, but left fibrations do since they are defined by the right lifting property with respect to subobjects of representables [11].…”

Section: Applicationsmentioning

confidence: 99%

“…Such a left fibration is not quite a fibration in a certain model category, but instead it defines a fibrant object in the covariant model structure on the over-category dSet/X. This model structure has been constructed in [11], where it is also shown to be Quillen equivalent to the model category of algebras (in sSet) over the simplicial operad associated to X.…”

Section: Introductionmentioning

confidence: 99%

“…We will explain and prove Proposition 1.2 in Section 4, where we also use the theory of minimal left fibrations to give an elementary proof of a result from [11] about weak equivalences between left fibrations.…”

Section: Introductionmentioning

confidence: 99%

“…o o which is a Quillen equivalence whenever f is an operadic weak equivalence ( [11], Proposition 2.4). As such, one obtains a (relative) functor…”

Section: Introductionmentioning

confidence: 99%

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Minimal fibrations of dendroidal sets

Moerdijk

Nuiten

2016

Algebr. Geom. Topol.

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Abstract. We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for ∞-operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. In an appendix, we explain how our arguments can be used to extend the results of [6], giving the existence of minimal fibrations in model categories of presheaves over generalised Reedy categories of a rather common type. Besides some applications to the theory of algebras over ∞-operads, we also prove a gluing result for parametrized connective spectra (or Γ-spaces).

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Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…o o which is a Quillen equivalence whenever f is an operadic weak equivalence ( [11], Proposition 2.4). As such, one obtains a (relative) functor…”

Section: Introductionmentioning

confidence: 99%

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Minimal fibrations of dendroidal sets

Moerdijk

Nuiten

2016

Algebr. Geom. Topol.

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An Introduction to Higher Categories

Paoli

2019

Algebra and Applications

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Lecture Notes on Infinity-Properads

Hackney

Robertson

2018

MATRIX Book Series

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The main goal of this lecture series is to provide a brief introduction to the theory of higher operads and properads. As these informal lecture notes stay very close to our presentations, which occupied only three hours in total, we were necessarily extremely selective in what is included. It is important to reiterate that this is not a survey paper on this area, and the reader will necessarily have to use other sources to get a 'big picture' overview.Various models of infinity-operads have been developed in work of C. Barwick, D.-C. Cisinski, J. Lurie, I. Moerdijk, I. Weiss and others [1,8,9,10,18,20,21]. In these lectures we focus on the combinatorial models which arise when one extends the simplicial category ∆ by a category of trees Ω. This 'dendroidal category' leads immediately to the category of dendroidal sets [20], namely the presheaf category Set Ω op . A dendroidal set X ∈ Set Ω op which satisfies an inner horn-filling condition is called a quasi-operad (see Definition 1.14). We briefly review these objects in section 1.Properads are a generalization of operads introduced by B. Vallette [23] which parametrize algebraic structures with several inputs and several outputs. These types of algebraic structures include Hopf algebras, Frobenius algebras and Lie bialgebras. In our monograph [12] with D. Yau and in subsequent papers, we work to generalize the theory of infinity-operads to the properad setting. In section 2 we explain the appropriate replacement of the dendroidal category Ω the graphical category Γ and define quasi-properads as graphical sets which satisfy an inner horn-filling condition. This material (and much more) can be found in the monograph [12]. It is worth mentioning that J. Kock, while reading the manuscript of [12], realized that one can give an alternative definition of the category Γ. The interested reader can find more details of this construction in [17].

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On the equivalence between Lurie's model and the dendroidal model for infinity-operads

Cited by 38 publications

References 32 publications

Minimal fibrations of dendroidal sets

Minimal fibrations of dendroidal sets

An Introduction to Higher Categories

Lecture Notes on Infinity-Properads

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