Dendroidal Sets

  title={Dendroidal Sets},
  author={Ieke Moerdijk and Ittay Weiss},
We introduce the concept of a dendroidal set. This is a generalization of the notion of a simplicial set, specially suited to the study of (coloured) operads in the context of homotopy theory. We define a category of trees, which extends the category  used in simplicial sets, whose presheaf category is the category of dendroidal sets. We show that there is a closed monoidal structure on dendroidal sets which is closely related to the Boardman–Vogt tensor product of (coloured) operads… Expand
C∗–algebraic drawings of dendroidal sets
In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a $C^*$-algebraic drawing of a dendroidal set. ItExpand
On the combinatorics of faces of trees and anodyne extensions of dendroidal sets
We discuss the combinatorics of faces of trees in the context of dendroidal sets and develop a systematic treatment of dendroidal anodyne extensions. As the main example and our motivation, we proveExpand
Homology of dendroidal sets
We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chainExpand
Dendroidal sets and simplicial operads
We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is aExpand
Dendroidal sets as models for homotopy operads
The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets isExpand
Dendroidal sets as models for connective spectra
Dendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on theExpand
Algebraic K-Theory of ∞-Operads
The theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal set D satisfying certain liftingExpand
Algebraic K-Theory of infinity-Operads
The theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads by Moerdijk and Weiss. An infinity-operad is a dendroidal set D satisfying certainExpand
Homology of infinity-operads
In a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of theExpand
Minimal fibrations of dendroidal sets
We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for infinity-operads, as well as in the covariant model structure for algebras and in the stableExpand


On quasi-categories as a foundation for higher algebraic stacks
We develop the basic theory of quasi-categories (a.k.a. weak Kan complexes or (oo, 1)categories as in [BV73], [Joy], [Lur06]) from first principles, i.e. without reference to model categories orExpand
Les Pr'efaisceaux comme mod`eles des types d''homotopie
Grothendieck introduced in Pursuing Stacks the notion of test category . These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes. A wellExpand
We deÞne natural A1-transformations and construct A1-category of A1-functors. The notion of non-strict units in an A1-category is introduced. The 2-category of (unital) A1-categories, (unital)Expand
Cyclic Operads and Cyclic Homology
The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the non-commutative setting. Recently, thereExpand
Koszul duality for Operads
(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] andExpand
We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointedExpand
Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory,Expand
Tensor product of operads and iterated loop spaces
Abstract A algebraic characterization of an n -fold loop space in terms of its n different 1-fold loop structures is established. This amounts to describing the higher homotopy commutativity for suchExpand
Higher Operads, Higher Categories
Part I. Background: 1. Classical categorical structures 2. Classical operads and multicategories 3. Notions of monoidal category Part II. Operads. 4. Generalized operads and multicategories: basicsExpand
Reports of the Midwest Category Seminar I
Hopf and Eilenberg-Maclane algebras.- Discoherently associative bifunctors on groups.- Directed colimits and sheaves in some non-abelian categories.- Bifibration induced adjoint pairs.- The doubleExpand