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

Gijs Heuts, Vladimir Hinich, Ieke Moerdijk

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)869-1043
Number of pages175
JournalAdvances in Mathematics
StatePublished - 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.


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

ASJC Scopus subject areas

  • Mathematics (all)


Dive into the research topics of 'On the equivalence between Lurie's model and the dendroidal model for infinity-operads'. Together they form a unique fingerprint.

Cite this