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

Gijs Heuts; Vladimir Hinich; Ieke Moerdijk

doi:10.1016/j.aim.2016.07.021

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

Gijs Heuts, Vladimir Hinich, Ieke Moerdijk

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We compare two approaches to the homotopy theory of ∞-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and ∞-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Γ. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.

Original language	English
Pages (from-to)	869-1043
Number of pages	175
Journal	Advances in Mathematics
Volume	302
DOIs	https://doi.org/10.1016/j.aim.2016.07.021
State	Published - 22 Oct 2016

Bibliographical note

Funding Information:
Parts of this paper were written while the second author was visiting the IHES, and the third author was visiting the Université de Paris VII, and the Newton Institute and St. John's College in Cambridge, respectively. We are grateful to these institutions for their hospitality and excellent working conditions. In addition, we would like to thank the Dutch Science Foundation (NWO) for supporting several mutual visits and the Académie des Sciences for supporting Moerdijk's visit to Paris through a Descartes-Huygens Prize. The second author acknowledges support from the Israel Science Foundation , grant # 446/15 . The authors would like to thank Jacob Lurie for several useful conversations and Luis Pereira for pointing out a mistake in an earlier version of the proof of Proposition 3.6.2 . Finally, we thank the anonymous referee for his or her careful reading of the manuscript.

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

Dendroidal sets
Forest sets
Infinity-operads
Quillen equivalence
Quillen model structures
Simplicial operads

ASJC Scopus subject areas

Mathematics (all)

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10.1016/j.aim.2016.07.021

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title = "On the equivalence between Lurie's model and the dendroidal model for infinity-operads",

abstract = "We compare two approaches to the homotopy theory of ∞-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and ∞-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Γ. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.",

keywords = "Dendroidal sets, Forest sets, Infinity-operads, Quillen equivalence, Quillen model structures, Simplicial operads",

author = "Gijs Heuts and Vladimir Hinich and Ieke Moerdijk",

note = "Funding Information: Parts of this paper were written while the second author was visiting the IHES, and the third author was visiting the Universit{\'e} de Paris VII, and the Newton Institute and St. John's College in Cambridge, respectively. We are grateful to these institutions for their hospitality and excellent working conditions. In addition, we would like to thank the Dutch Science Foundation (NWO) for supporting several mutual visits and the Acad{\'e}mie des Sciences for supporting Moerdijk's visit to Paris through a Descartes-Huygens Prize. The second author acknowledges support from the Israel Science Foundation , grant # 446/15 . The authors would like to thank Jacob Lurie for several useful conversations and Luis Pereira for pointing out a mistake in an earlier version of the proof of Proposition 3.6.2 . Finally, we thank the anonymous referee for his or her careful reading of the manuscript. Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

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KW - Forest sets

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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