Operads and PROPs

Algebras over infinity-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…

Gröbner Bases for Coloured Operads

In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Gröbner bases for coloured operads. Among the examples for which we show the existance of a quadratic Gröbner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).

Derived A-infinity algebras in an operadic context

Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show Sagave's definition of morphisms agrees…

拟模操作子

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。

Protoperads i: Combinatorics and definitions

This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced S-modules. We study the associated monoids, called protoperads, which are a generalization of operads. As operads encode algebraic operations with several inputs and one outputs, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and show that there exists a notion of free protoperad. We also show that the monoidal product introduced here is related t…

弯曲operad与弯曲代数的同伦论

Homotopy theory of curved operads and curved algebras

本文通过弯曲operad研究弯曲代数，发展bar与cobar构造及Koszul对偶理论，在过滤完备对象上定义组合模型范畴结构，并证明弯曲结合代数与弯曲A∞-代数的同伦范畴Quillen等价。

Forests and the W Construction

A modified definition of the category of forests by Costello and expounded upon by Getzler is given. This language is used to prove some cofibration properties of the W construction for colored operads, following the work of Berger-Moerdijk. An application is given to the swiss cheese operad. The language of forests is convenient for showing that the swiss cheese operad is equivalent to its free degree 0 and 1 pieces.

Manin products, Koszul duality, Loday algebras and Deligne conjecture

In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these opera…

The mod 2 homology of free spectral Lie algebras

The Goodwillie derivatives of the identity functor on pointed spaces form an operad in spectra. Adapting a definition of Behrens, we introduce mod 2 homology operations for algebras over this operad and prove these operations account for all the mod 2 homology of free algebras on suspension spectra of simply-connected spaces.

Iterated bar complexes of E-infinity algebras and homology theories

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to…

控制 props 与 wheeled props 的 Koszul operads

Koszul Operads Governing Props and Wheeled Props

本文构造了控制 props 和 wheeled props 的群胚着色 operads，证明其 Koszul 性，并由此定义同伦 props、构造 Massey 积以刻画形变性，重获 Mac Lane 定理。

Algebras of higher operads as enriched categories II

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimb…

Iterated suspensions are coalgebras over the little disks operad

We study the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every $n$-fold suspension is a coalgebra over the little $n$-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an $n$-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an $n$-fold suspension to be an $(n+1)$-fold suspension, i.e. if this cooperation is non-zero then the space is not an $(n+1)$-fold suspension. We prove several results in equivaria…

On PreLie algebras with divided symmetries

We study an analogue of the notion of p-restricted Lie-algebra and of the notion of divided power algebra for PreLie-algebras. We deduce our definitions from the general theory of operads. We consider two variants Λ(P,-) and Γ(P,-) of the monad S(P,-) which governs the category of algebras classically associated to an operad P. For the operad of PreLie-algebras P=PreLie, we prove that the category of algebras over the monad Λ(PreLie,-) is identified with an already defined category of p-restricted PreLie-algebras introduced by A. Dzhumadil'daev. We give an explicit description of the structure…

带除幂的同伦预李代数与operad映射空间的同伦

Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces

本研究定义了带除幂的同伦预李代数相关的$ΓΛ\mathcal{PL}_\infty$-代数与Maurer-Cartan元，证明了相关代数结构与同伦不变性，推广了Goldman-Millson定理，并用于刻画operad范畴的映射空间。

Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equival…

Profinite $\infty$-operads

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model category of profinite infinity-operads. The construction is based on a notion of lean infinity-operad, and we characterize those infinity-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) infinity-operads and of infinity-categories…

Operad groups and their finiteness properties

We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.

Gerstenhaber algebra and Deligne's conjecture on Tate-Hochschild cohomology

Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…

操作范畴的Koszul对偶

Koszul duality for operadic categories

本文为一大类操作范畴上的operads建立Koszul对偶与Koszul性理论，研究二元二次operads实例并证明其Koszul性，涵盖经典operads、cyclic、modular、wheeled operads及多种PROPs变体等结构。

Symmetrisation of $n$-operads and compactification of real configuration spaces

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint $Sym_1$ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmmetric operads are 1-operads and $Sym_1$ is the first term of the infinite series of left adjoint functors $Sym_n,$ called symmetrisation functors, from $n$-operads to symmetric operads with the property that the category of one object, one arrow, . . ., one $…

Operadic multiplications in equivariant spectra, norms, and transfers

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N-infinity operads, equivariant generalizations of E-infinity operads. Algebras in equivariant spectra over an N-infinity operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N-infinity operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant in…

On the derived category of an algebra over an operad

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

Homotopy theory of non-symmetric operads II: change of base category and left properness

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.

Dwyer-Kan Homotopy Theory of Algebras over Operadic Collections

Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of enriched wheeled props, wheeled properads, and wheeled operads, among others. This result extends known model category structure on the categories of operads, properads, and props enriched in simplicial sets and other monoidal model categories. We also show that our Dwyer-Kan model category structure is well behaved with respect to simultaneous changes of the un…

Equivalence of Operads over Symmetric Monoidal Categories

In this paper, we study conditions for extending Quillen model category properties , between two symmetric monoidal categories, to their associated category of symmetric sequences and of operads. Given a Quillen equivalence $λ: \mathcal{C}=Ch_{\Bbbk,t} \leftrightarrows \mathcal{D}: R,$ so that $\mathcal{D}$ is any symmetric monoidal category and the adjoint pair $(λ, R)$ is weak monoidal, we prove that the categories of connected operads $Op_\mathcal{C}$ and $Op_\mathcal{D}$ are Quillen equivalent. This expands an analogous result of Schwede-Shipley(\cite{SS03}) when we replace these categorie…

针对同伦凝聚余代数的点集模型

Point-set models for homotopy coherent coalgebras

本文通过归纳法证明，当操作子为cofibrant时，通过局部化微分格子代数得到的∞-范畴与基于富集∞-操作子的余代数∞-范畴等价，从而为E_n和E_∞余代数提供显式点集模型。

连通乘法operad上的单纯结构

Simplicial Structure on Connected Multiplicative Operads

本研究在连通乘法operad上定义新的单纯结构，引入右brace代数结构得到双复形，定义两种乘积并结合微分算子得到两类微分分次代数，还通过Alexander-Withney映射得到微分分次余代数，最后给出实例。

Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.

$\u211e_\u221e$ 操作子的稳定化与 $p$ 进稳定同伦论

Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory

该研究构造了作为 $\u211e_\u221e$ 操作子稳定化版本的稳定操作子，建立其代数同伦与上同调运算理论，并证明谱上链赋予其结构后可作为 $p$ 进稳定同伦型的代数模型。

作为Koszul对偶自然环境的操作范畴

Operadic categories as a natural environment for Koszul duality

本文为操作范畴上的operads建立Koszul对偶理论框架，提炼出支持二次operads及其Koszul性发展的附加性质，并利用离散操作（op）纤维化构造新操作范畴，重点证明了图的操作范畴满足全部附加性质。

相对 operad 与 PROP 的特征标 plethysm

Plethysm for characters of relative operads and props

本文将对称函数特征标理论推广至相对 operad 与 PROP，提出相对 plethysm 和连通盒积以刻画相应复合结构，并用于计算自由群及 IA-自同构群上同调的特征标。

Inverting operations in operads

We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization. For an operad O and a sub-monoid of one-ary operations W we associate an operad LO and a canonical map O to LO which takes elements in W to homotopy invertible operations. Furthermore, we give a functor from the category of O-algebras to the category of LO-algebras satisfying an appropriate universal property.

André-Quillen cohomology of algebras over an operad

We study the André-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated André-Quillen cohomology equal to an Ext-functor ? We give several criterion, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads.

Modular operads as modules over the Brauer properad

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

The higher Morita category of $E_n$-algebras

We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a monoidal $\infty$-category. By working with $\infty$-operads over $Δ^{n,\text{op}}$ we iterate these definitions and generalize our construction to get an $(\infty,n+1)$-category of $E_{n}$-algebras and iterated bimodules in an $E_{n}$-monoidal $\infty$-category. Moreover, we show that if $\mathcal{C}$ is an $E_{n+k}$-monoidal $\infty$-category then the $(\infty…

The Exterior Graded Swiss-Cheese Operad $Λ^{S^2}_V$ (with an appendix by Ana Lorena Gherman and Mihai D. Staic)

In this paper we present a construction which is a generalization of the exterior algebra of a vector space $V$. We show how this fits in the language of operads, discuss some properties, and give explicit computations for the case $dim(V)=2$.

Weyl n-algebras and the Swiss cheese operad

We apply Weyl $n$-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

operadic 代数的霍奇schild 复形与余切复形

Hochschild and cotangent complexes of operadic algebras

该研究利用Lurie的余切复形形式化构建富集operad上代数的Quillen上同调，引入谱霍奇schild上同调理论，证明两种上同调可由operad自身控制，在单纯集范畴中可通过谱值函子计算，并建立Eₙ-空间两种复形的上纤维序列及非稳定类似物。

小立方体运算符的普遍性质

Universal Properties of Variations of the Little Cubes Operads

本文研究了小立方体运算符的变体的普遍性质，展示了一个运算符映射等价于特定的等变映射，从而显式地将运算符表示为一个列限。

Symmetric homotopy theory for operads

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provi…

代数operad上代数的高阶Massey积

Higher order Massey products for algebras over algebraic operads

本文为代数operad上的代数引入高阶Massey积，推广了Muro的次级积，研究其性质、形式性联系，并证明其表示operadic Eilenberg--Moore谱序列的微分，同时探讨与P∞结构的关系，涵盖Koszul及非Koszul情形。

Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this first part, we set up th…

From operator categories to topological operads

In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category $Λ(Φ)$ attached to a perfect operator category $Φ$ that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman--Vogt tensor product that lies over it. We then give examples of operator categories that provid…

对运算子和代数的模型结构从全局视角研究

Model structures on operads and algebras from a global perspective

本文从全局视角研究运算子和其代数、模的同伦理论，通过模型范畴和半模型范畴提供统一框架，并在新环境中产生模型结构、校正结果和适当性结果。

基于子图收缩的算子代数的一种推广

A generalization of operads based on subgraph contractions

本文提出算子代数的推广——收缩代数，其操作集由连通图索引、复合规则由连通子图收缩编号，证明了多种经典算子代数可推广至该框架，并验证了Koszul对偶与Gröbner基等工具的适用性及特定收缩代数的Koszul性质。