Peter Bonventre, Luis Alexandre Pereira
We give an explicit description of the rigidification of an $\infty$-operad as a simplicial operad. This description is based on the notion of dendroidal necklace, extending work of Dugger and Spivak from the categorical context to the operadic context, although with a different framework, which relates constructions involving necklaces to a standard factorization of maps in the category of trees.
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We discuss a variant of the category of dendroidal sets, the so-called closed dendroidal sets which are indexed by trees without leaves. This category carries a Quillen model structure which behaves better than the one on general dendroidal sets, mainly because it satisfies the pushout-product property, hence induces a symmetric monoidal structure on its homotopy category. We also study complete Segal style model structures on closed dendroidal spaces, and various Quillen adjunctions to model categories on (all) dendroidal sets or spaces. As a consequence, we deduce a Quillen equivalence from …
We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-$\infty$-operads in equivariant dendroidal sets.
Dendroidal sets offer a formalism for the study of $\infty$-operads akin to the formalism of $\infty$-categories by means of simplicial sets. We present here an account of the current state of the theory while placing it in the context of the ideas that lead to the conception of dendroidal sets. We briefly illustrate how the added flexibility embodied in $\infty$-operads can be used in the study of $A_{\infty}$-spaces and weak $n$-categories in a way that cannot be realized using strict operads.
We introduce the analogues of the notions of complete Segal space and of Segal category in the context of equivariant operads with norm maps, and build model categories with these as the fibrant objects. We then show that these model categories are Quillen equivalent to each other and to the model category for $G$-$\infty$-operads built in a previous paper. Moreover, we establish variants of these results for the Blumberg-Hill indexing systems. In an appendix, we discuss Reedy categories in the equivariant context.
A dendroidal approach to operadic right modules and manifold calculus
本文通过森林空间形式研究运算子右模的同伦论，证明在满足条件的运算子下，项目模型结构与切类范畴的反变模型结构存在Quillen等价，用于简化导出映射空间计算及分析Goodwillie-Weiss塔结构。
On the equivalence of the Lurie's $\infty$-operads and dendroidal $\infty$-operads
本研究聚焦Lurie定义的$\infty$-operads与基于树状空间的树状$\infty$-operads的对称幺半$\infty$-范畴，证明二者等价，完善了高阶代数中operad理论的相关内容。
We extend the Cisinski-Moerdijk-Weiss theory of $\infty$-operads to the equivariant setting to obtain a notion of $G$-$\infty$-operads that encode "equivariant operads with norm maps" up to homotopy. At the root of this work is the identification of a suitable category of $G$-trees together with a notion of $G$-inner horns capable of encoding the compositions of norm maps. Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work.
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 is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.
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 infinity-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 …
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 certain lifting conditions. In this paper we give a definition of K-groups K_n(D) for a dendroidal set D. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. We show that t…
We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-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 Gamma. 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.
A new model structure on dendroidal spaces for the theory of $\infty$-operads
本文提出了一种新的 dendroidal 空间模型结构，用于 $\infty$-操作符理论，并证明其与 Cisinski-Moerdijk 模型结构的 Quillen 等价性。
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 operads in the context of homotopy theory. We define a category of trees, which extends the category $Δ$ 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 operads. Furthermore we show that each operad in a suitable model category has a coherent homotopy nerve which is a dendroidal set, extend…
We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for infinity-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal's quasi-categories, Rezk's complete Segal spaces, and…
The right cancellation property for certain classes of dendroidal anodynes
本文将Stevenson的结果推广至树状集合范畴，证明了树状内单形扩张在正规单态射类中的右消去性质，并以此为基础，从树状∞-operad构造出对称幺半∞-范畴。
Relative dendroidal Rezk nerve and applications
将树状Rezk神经推广到相对∞-operads，通过推广Mazel-Gee定理建立与∞-operads局部化的关系，并应用于operadic局部化，得到包括Willwacher结果推广和球面上局部常值因子代数离散几何描述等新结果。
Models for cyclic infinity operads
本文在循环树状集与空间上构建了循环拟算畴与完备循环树状 Segal 空间的模型结构，证明其与单纯循环算畴 Quillen 等价，并推广至平面情形以支持 Walde 纲领。
We show that normalized cacti form an $\infty$-operad in the form of a dendroidal space satisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketed trees and an enrichment of the dendroidal category $Ω$.
We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.
We develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets and extends Lurie's definition of a coCartesian fibration of simplicial sets. We show how, for a dendroidal set X, the coCartesian fibrations over X fit together to form an infinity-category coCart(X). Using a generalization of the Grothendieck construction, we prove that coCart(X) is equivalent to the infinity-category of algebras in infinity-categories ov…
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 stable one for connective spectra. In an appendix, we explain how our arguments can be used to extend the results of Cisinski, 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 infinity-operads, we also prove a gluing result for parametrized connective spectra (or…
We exhibit the simplex category $Δ$ and Segal's category $Γ$ as $\infty$-categorical localizations of the dendroidal categories $Ω_π$ and $Ω$ introduced by Moerdijk and Weiss. As an application we obtain an equivalence of $\infty$-categories between invertible $\infty$-operads and the $2$-Segal spaces of Dyckerhoff and Kapranov. Finally, we describe a cyclic version of the dendroidal category and explain how it $\infty$-localizes to Connes's cyclic category $Λ$.
In our paper "Dendroidal sets as models for homotopy operads" (J. Topol. 4 (2011), no. 2, 257-299, and arXiv:0902.1954), we made the wrong claim about the behaviour of the tensor product with respect to cofibrations of dendroidal sets. We added an erratum at the end of the arXiv version of loc. cit. This short note contains the proof of a technical lemma used in the erratum.
The category of dendroidal sets is an extension of that of simplicial sets, suitable for defining nerves of operads rather than just of categories. In this paper, we prove some basic properties of inner Kan complexes in the category of dendroidal sets. In particular, we extend fundamental results of Boardman and Vogt, of Cordier and Porter, and of Joyal to dendroidal sets.
We propose a construction of the monoidal envelope of $\infty$-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces.
We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give an Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.
Segal conditions for generalized operads
本文介绍Moerdijk--Weiss dendroidal sets的推广，通过图范畴预层的Segal条件识别广义operad，并利用左Kan延拓实现自由/遗忘伴随，涉及同伦相干版本。
We develop an $\infty$-categorical version of the classical theory of polynomial and analytic functors, initial algebras, and free monads. Using this machinery, we provide a new model for $\infty$-operads, namely $\infty$-operads as analytic monads. We justify this definition by proving that the $\infty$-category of analytic monads is equivalent to that of dendroidal Segal spaces, known to be equivalent to the other existing models for $\infty$-operads.
This paper describes a consequence of the more general results of a previous paper which is of independent interest. We construct a functor from the category of dendroidal sets, which models the theory of infinity-operads, into the category of E-infinity-spaces. Applying May's infinite loop space machine for E-infinity-spaces then gives an infinite loop space machine for infinity-operads. We show that our machine exhibits the homotopy theory of E-infinity-spaces as a localization of the homotopy theory of infinity-operads.
Quasi Modular Operads
本文定义了无穷模操作子的Kan条件，证明了神经定理及Kan与Segal条件的等价性，还在附录中补充了循环操作子的相关对应内容。
We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Ou…
In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice sym…
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 the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define an adjoint bar-cobar (or "Koszul") duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of tree…
The root functor
本文证明任何∞-operad等价于离散Σ-free operad的局部化，将Joyal范畴去局部化定理推广到operad情形，并系统研究树状结构中的∞-operadic局部化及其与直/非直化等价的兼容性。
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
Unital k-Restricted Infinity-Operads
本文通过元数限制研究单位无穷操作范畴，建立单位 k-限制无穷操作范畴的完全 Segal 预层模型，并证明限制函子有完全忠实的左右伴随，从而给出过滤与余过滤结构。
A straightening-unstraightening equivalence for $\infty$-operads
本文为 $\infty$-operads 建立伸直-非伸直伴随等价，并证明 Hinich-Moerdijk 比较函子诱导 operadic 与 dendroidal 左纤维化等价，同时刻画幺半非伸直函子的本质像。
We study the (\infty, 1)-category of autoequivalences of \infty-operads. Using techniques introduced by Toën, Lurie, and Barwick and Schommer-Pries, we prove that this (\infty, 1)-category is a contractible \infty-groupoid. Our calculation is based on the model of complete dendroidal Segal spaces introduced by Cisinski and Moerdijk. Similarly, we prove that the (\infty, 1)-category of autoequivalences of non-symmetric \infty-operads is the discrete monoidal category associated to Z/2Z. We also include a computation of the (\infty, 1)-category of autoequivalences of (\infty, n)-categories based…
These are notes for three lectures on higher properads given at a program at the mathematical institute MATRIX in Australia in June 2016. The first lecture covers the case of operads, and provides a brief introduction to the Moerdijk-Weiss theory of dendroidal sets. The second lecture extends the discussion to properads and our work with Donald Yau on graphical sets. These two lectures conclude with models for higher (pr)operads given by an inner horn filling condition. Finally, in the last lecture, we explore some properties of the graphical category and use them to give a Segal-type model fo…
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