Michael Batanin, Clemens Berger
We introduce two coloured operads in sets -- the lattice path operad and a cyclic extension of it -- closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an $E_2$-action (resp. framed $E_2$-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.
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In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them.
Combinatorial operad actions on cochains
通过对称群的棒构造，形成了一个经典的E-无穷运算。M. Barratt和P. Eccles在单纯集的上下文中，以便有一个类似的Milnor FK-建设无限闭路空间。证明了单纯集的正规上链复形上的结合代数结构推广到Barratt-Eccles运算上的代数结构.我们还证明了Barratt-Eccles算子上的微分分次代数构成一个闭模型范畴。类似的结果也适用于结合代数的正规化Hochschild上链复形。更确切地说，Hochschild上链复形是由Barratt-Eccles操作数的一个子操作数作用的，它等价于经典的小平方操作数。
We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.
We construct a Frobenius algebra structure on the Hochschild cochains of a group ring k[G] that extends the known structure of a <1, 2> topological quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The convolution product extends to the homotopy commutative Gerstenhaber product on cochains, the Frobenius coproduct extends to a coproduct on the chain complex for Hochschild homology, and there is a pairing on Hochschild cocahins satisfying Frobenius associativity. The pairing, however, degenerates on a certain subcomplex of Hochschild cochains. The cochain complex for gro…
We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra.
We study a naturally occurring $E_{\infty}$-subalgebra of the full $E_2$-Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial cohomology of the classifying space of the underlying group or category, $\cal{C}$. In this setting the simplicial cup product of cochains on $B{\cal{C}}$ agrees with the Gerstenhaber product and Steenrod's cup-one product of cochains agrees with the pre-Lie product. We extend the idea of coherent cochains to algebras more general than category algebras and dub …
This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in arXiv:1605.00842. This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is p…
Plus-construction for algebras over an operad, cyclic and Hochschild homologies up to homotopy
本文将同伦局部化应用到算子上的微分分次代数的框架中。通过对泛无圈代数进行零化得到了加号的构造。李代数和莱布尼兹代数到同伦的这一加法构造给出了结合代数的两个新的同调论。它们与循环同调和霍赫希尔德同调密切相关。我们还计算了它们的第一同调群。
We transport Steenrod's cup-i products from the singular cochains on the free loop space Maps(S^1, BG) to Hochschild's original cochain complex Hom (k[G]^*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary coefficient ring, and BG the classifying space of G. For cochains supported on BG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocycles f and g supported on BG, the Gerstenhaber bracket [f, g] = 0 in Hochschild cohomology. This i…
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。
Formality of $\mathbb{E}_n$-algebras and cochains on spheres
研究增广𝔼ₙ-代数的环路与悬垂函子，证明n-球面的上链代数作为𝔼ₙ-代数是形式的，且谱中操作子到单子的自由函子在含稳定𝔼ₙ-操作子的子范畴上完全忠实。
In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichmüller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite…
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…
On the Multiplicative Structure of Topological Hochschild Homology
证明了E_n-环谱R的拓扑霍赫希尔德同调THH（R）是E_{n-1} -环谱.证明是基于这样一个事实，即张量积的操作数的A SS幺半群结构和小n-立方体操作数是一个E_{n+1} -操作数，结果是独立的兴趣。
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。
The Hochschild cohomology of a Poincaré algebra
本文定义了仙人掌集的概念，并证明了它的几何实现是Voronov仙人掌操作数上的代数，它等价于框架二维小圆盘操作数$\mathcal{D}_2$ .利用这一点，我们证明了Poincaré代数A的Hochschild上同调是$\mathcal{D}_2$（的链复形）上的代数。
Multivariable cochain operations and little n-cubes
这是对2001年6月4日首次发表的一篇论文的修订。它将出现在AMS杂志上。本文构造了一个自然作用于拓扑空间的正规上链$S^*X$的小$E_\infty$链运算$§$.对于每个$n$，我们还构造了一个子运算子$§_n$，它与小$n$ -立方运算子的归一化奇异链是准同构的. $n=2$的情况下，导致我们的Deligne的Hochschild上同调猜想的早期证明大大简化。
If P is a dg-operad acting on a dg-algebra A via algebra homomorphisms, then P acts on the Hochschild complex of A. In the more general case when P is a dg-prop, we show that P still acts on the Hochschild complex, but only up to coherent homotopy. We moreover give a functorial dg-replacement of P that strictifies the action. As an application, we obtain an explicit strictification of the homotopy coherent commutative Hopf algebra structure on the Hochschild complex of a commutative Hopf algebra.
The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous results to prove that multiplicative 1-operads in duoidal categories admit, under some mild conditions on the underlying monoidal category, natural actions of contractible 2-operads. The result of D. Tamarkin on the structure of dg-categories, as well as the classical Deligne conjecture for the Hochschild cohomology, is a particular case of this statement.
Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads
这是两篇论文中的第一篇，我们证明了细胞模型的模空间曲线的标记点和切向量在标记点上的Hochschild余链的Frobenius代数。我们还证明了有dg-PROP作用的沙利文弦图的一个版本，其作用于Frobenius代数的规范化Hochschild余链。这些动作提升到余圈上的运算相关函数。特别地，PROP作用给出了关于紧致单连通流形的闭路空间的同调的作用。在第一部分中，我们建立了拓扑操作/PROPs及其单元模型。这一部分的主要定理是：具有$n$穿孔和在每个穿孔处的切向量的亏格$g$曲线的模空间存在胞模操作;存在CW复形，其链同构于某种类型的Sullivan Chord图，并且它们构成一个PRO;并且在拓扑水平上存在这些结构的弱形式，它们都位于一个全拓扑平面内包含循环（有理数）操作数。
Deformations of algebras over operads and Deligne's conjecture
本文发展了算子和算子上代数的变形理论。自由分解（通过Boardman-Vogt方法构造）用于描述变形的形式模空间。我们应用一般理论证明了Deligne猜想。后者认为结合代数的Hochschild复形在小圆盘操作数的链操作数上具有dg-代数的标准结构。在证明的过程中，我们构建了一个操作的几何性质的Hochschild复杂的行为。它似乎不同于支撑操作数（后者用于以前的方法德利涅猜想）。从我们的结果可以得出，Grothendieck-Teichmüller群（同伦）作用在Hochschild复形上的2-代数的结构的模空间上。在附录中，我们发展了一个理论的分段代数链和形式。它适用于带角的实半代数流形（如点的配置空间的富尔顿-麦克弗森紧化）。
Brace algebras and the cohomology comparison theorem
Gerstenhaber和Schack的上同调比较定理证明了一个代数的Hochschild复形与一个空间的奇异上链复形之间存在上链等价。我们表明，这个比较定理保持支撑代数结构。这个结果给出了最近的结果建立精细的拓扑结构的Hochschild上同调的结构原因，和一个简单的方法来获得它们从相应的性质的上链复形。
Operations on the Hochschild Bicomplex of a Diagram of Algebras
本文构造了一个作用在代数图 Hochschild 双复形上的 operad，由此直接证明其 Hochschild 上同调为 Gerstenhaber 代数，全复形为 $L_\infty$-代数，并通过 Maurer-Cartan 方程控制形变。
Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators
这是两篇论文中的第二篇，我们证明了一个细胞模型的模空间的曲线与标记点和切向量在标记点上的Hochschild余链的Frobenius代数。我们还证明了有dg-PROP作用的沙利文弦图的一个版本，其作用于Frobenius代数的规范化Hochschild余链。这些动作提升到余圈上的运算相关函数。特别地，PROP作用给出了关于紧致单连通流形的闭路空间的同调的作用。在这第二部分中，我们离散的第一部分的操作和PROPic结构。我们还介绍了运算相关函数的概念，并使用它们与运算映射从细胞水平的离散化的对象来定义相关的行动。
Posets, their Incidence Algebras and Relative Operads, and the Cohomology Comparison Theorem
该研究深入探究 Gerstenhaber-Schack 上同调比较定理，证明其保持偏序集关联代数相关复形上的多种代数结构，为偏序集与拓扑学提供新联系。
In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. Moreover, we prove a generalization to the equivariant context.
This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our …
A solution of Deligne's conjecture
Deligne在1993年问是否Hochschild上链复形的结合环有一个自然的行动的奇异链的小2立方运算。本文对此问题给予肯定的回答。我们还证明了结合环谱的拓扑Hochschild上同调谱具有等价于小2-立方的运算作用。
A combinatorial approach to Kontsevich's Swiss cheese conjecture
本文通过构造新的操作算子SC(P)和Hochschild对象，证明了当P为小n-盘操作算子时，SC(P)等价于瑞士奶酪操作算子SC_n，并应用此结果证明了交换代数的n阶Hochschild-Pirashvili余链上存在E_{n+1}-作用。
For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss Cheese version of the Deligne conjecture formulated by M. Kontsevich in arXiv:math/9904055.
Brace operations and Deligne's Conjecture for module-algebras
本文证明了模代数的Kaygun Hopf-Hochschild上链复形是一个带乘法的支撑代数。因此，（i）Deligne猜想的类似物对模代数成立，（ii）模代数的Hopf-Hochschild上同调具有Gerstenhaber代数结构。
A proof of a cyclic version of Deligne's conjecture via Cacti
本文证明了具有非退化、对称、不变内积的结合代数的正规化Hochschild余链是Cacti给出的框架圆盘操作链模型上的代数.特别是，在这个意义上，它们是BV代数的同伦和Hochschild上同调这样的代数是BV代数的诱导括号与Gerstenhaber的括号重合。为了证明这一点，我们使用一个细胞链模型的框架小光盘操作规范化仙人掌。这个模型是通过张紧我们的链模型的小光盘operad在无刺仙人掌与自然链模型$(S^1)^{\times n}$适应仙人掌。
We explain our previous results about Hochschild actions [Kau07a, Kau08a] pertaining in particular to the coproduct which appeared in a different form in [GH09] and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in [Kau08a], is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations of [Kau07a, Kau08a]. With respect to loop spaces, w…
The cyclic Deligne conjecture and Calabi-Yau structures
本文证明了循环Deligne猜想，指出配备Calabi-Yau结构的线性范畴的Hochschild上链具有framed $E_2$-代数结构，并应用于相对环路同调上的弦拓扑运算。
The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a hom…
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