Dendroidal Segal spaces and ∞-operads

Cisinski, Denis-Charles; Moerdijk, Ieke

doi:10.1112/jtopol/jtt004

Cited by 68 publications

(93 citation statements)

References 17 publications

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“…and hcN d of the main theorem form a Quillen pair (Proposition 4.9), and we prove that the square ( * ) is commutative in a homotopy theoretic sense, even though the top horizontal functor is not a right-Quillen functor (although it does preserve weak equivalences). Thus, using the fact, from our earlier paper [10], that the inclusion functors relating dSet, sdSet and PreOper are left-Quillen equivalences, we see that, to prove that W ! and hcN d form a Quillen equivalence, it is in fact enough to prove that the adjunction between preoperads and simplicial operads is a Quillen equivalence.…”

Section: Introductionmentioning

confidence: 84%

“…The category of dendroidal sets is a category of presheaves of sets. In our earlier paper [10], we studied the related category of presheaves of simplicial sets: this is the category of dendroidal spaces, identical to the category of simplicial objects in dSet, and denoted sdSet. It contains as a full subcategory the category of preoperads, those dendroidal spaces whose space of vertices (or objects, or colours) is discrete.…”

Section: Introductionmentioning

confidence: 99%

“…It contains as a full subcategory the category of preoperads, those dendroidal spaces whose space of vertices (or objects, or colours) is discrete. In [10], we proved that the category of dendroidal spaces carries a Rezk style Quillen model structure whose fibrant objects are referred to as dendroidal complete Segal spaces. We established a Quillen equivalence between the original model category of dendroidal sets and this model category of dendroidal complete Segal spaces.…”

Section: Introductionmentioning

confidence: 99%

“…and hcN d form a Quillen equivalence, it is in fact enough to prove that the adjunction between preoperads and simplicial operads is a Quillen equivalence. In order to do this, we change the model structure on PreOper used in [10] into a 'tame' model structure, with the same weak equivalences but considerably fewer cofibrant objects (Section 7). With fewer cofibrant objects to deal with, it is possible to show that the functor N d on top of diagram ( * ) is a right-Quillen equivalence (Theorem 8.4).…”

Section: Introductionmentioning

confidence: 99%

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Dendroidal sets and simplicial operads

Cisinski

Moerdijk

2013

Journal of Topology

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127

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Abstract. 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 a right Quillen equivalence from the model category of simplicial operads to the model category structure for ∞-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dendroidal sets and simplicial operads

Cisinski

Moerdijk

2013

Journal of Topology

Self Cite

127

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“…Recent work of Cisinski and Moerdijk [11][12][13] and Heuts, Hinich and Moerdijk [17] relates the dendroidal sets world with that of Lurie. We hope that the methods presented in our paper will be of help in understanding the precise relation of their tensor products with the interchange of structures.…”

Section: Introductionmentioning

confidence: 94%

An Additivity Theorem for the interchange of En structures

Fiedorowicz

Vogt

2015

Advances in Mathematics

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Communicated by the Managing Editors of AIM Dedicated to the memory of Roland Schwänzl MSC: 55P48 55P35 18D50 Keywords: Operads Loop spaces Higher homotopy commutativity Interchange of structuresLet A and B be operads and let X be an object with an A-algebra and a B-algebra structure. These structures are said to interchange if each operation α : X n → X of the A-structure is a homomorphism with respect to the B-structure and vice versa. In this case the combined structure is codified by the tensor product A ⊗ B of the two operads. There is not much known about A ⊗ B in general, because the analysis of the tensor product requires the solution of a tricky word problem. Intuitively one might expect that the tensor product of an E k -operad with an E l -operad (which encode the multiplicative structures of k-fold, respectively l-fold loop spaces) ought to be an E k+l -operad. However, there are easy counterexamples to this naive conjecture. In this paper we essentially solve the word problem for the nullary, unary, and binary operations of the tensor product of arbitrary topological operads and show that the tensor product of a cofibrant E k -operad with a cofibrant E l -operad is an E k+l -operad. It follows that if A i are E ki operads for i = 1, 2, . . . , n, then A 1 ⊗. . .⊗A n is at least an E k1+...+kn operad, i.e. there is an E k1+...+kn -operad C and a map of operads C → A 1 ⊗ . . . ⊗ A n .

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Infinity Properads and Infinity Wheeled Properads

Hackney

Robertson

Yau

2015

Lecture Notes in Mathematics

180

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PrefaceThis monograph fits in the intersection of two long and intertwined stories. The first part of our story starts in the mid-twentieth century, when it became clear that a new conceptual framework was necessary for the study of higher homotopical structures arising in algebraic topology. Some better known examples of these higher homotopical structures appear in work of J. F. Adams and S. Mac Lane [Mac65] on the coproduct in the bar construction and work of J. Stasheff [Sta63], J. M. Boardman and R. M. Vogt [BV68,BV73], and J. P. May [May72] on recognition principles for (ordinary, n-fold, or infinite) loop spaces. The notions of 'operad' and 'prop' were precisely formulated for the purpose of this work; the former is suitable for modeling algebraic or coalgebraic structures, while the latter is also capable of modeling bialgebraic structures, such as Hopf algebras. Operads came to prominence in other areas of mathematics beginning in the 1990s (but see, e.g. This renaissance in the world of operads [Lod96, LSV97] and the popularity of quantum groups [Dri83,Dri87] in the 1990s lead to a resurgence of interest in props, which had long been in the shadow of their little singleoutput nephew. Properads were invented around this time, during an effort of B. Vallette to formulate a Koszul duality for props [Val07]. Properads and props both model algebraic structures with several inputs and outputs, but properads govern a smaller class of such structures, those whose generating operations and relations among operations can be taken to be connected. This class includes most types of bialgebras that arise in nature, such as biassociative bialgebras, (co)module bialgebras, Lie bialgebras, and Hopf algebras. The use of 'colored' or 'multisorted' variants of operads or props [Bri00, BV73], where composition is only partially defined, allows one to address many other situations of interest. It allows one, for instance, to model morphisms of algebras associated to a given operad. There is a two-colored operad which encodes the data of two associative algebras as well as a map from one to the other; a resolution of this operad precisely gives the correct notion of morphism of A ∞ -algebras [Mar04,Mar02]. It also provides a unified way to treat operads, cyclic operads, modular operads, properads, and so on: for each there is a colored operad which controls the structure in question.The second part of our story is an extension of the theory of categories. Categories are pervasive in pure mathematics and, for our purposes, can be loosely described as tools for studying collections of objects up to isomorphism and comparisons between collections of objects up to isomorphism. When the objects we want to study have a homotopy theory, we need to generalize category theory to identify two objects which are, while possibly not isomorphic in the categorical sense, equivalent up to homotopy. For example, when discussing topological spaces we might replace 'homeomorphism' with 'homotopy equivalence'. This leads to the theory of ∞-ca...

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Dendroidal Segal spaces and ∞-operads

Cited by 68 publications

References 17 publications

Dendroidal sets and simplicial operads

Dendroidal sets and simplicial operads

An Additivity Theorem for the interchange of En structures

Infinity Properads and Infinity Wheeled Properads

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