Paolo Salvatore
Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2-discs operad.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action of an operad equivalent to the little 2-cubes.
We give a topological solution to the $\Ainf$ Deligne conjecture using associahedra and cyclohedra. For this we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally acts on the Hochschild cochains of an $\Ainf$ algebra yielding an explicit, topological and minimal solution to the $\Ainf$ Delign…
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…
In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the l…
Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.
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 aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
It is observed that Kaygun's Hopf-Hochschild cochain complex for a module-algebra is a brace algebra with multiplication. As a result, (i) an analogue of Deligne's Conjecture holds for module-algebras, and (ii) the Hopf-Hochschild cohomology of a module-algebra has a Gerstenhaber algebra structure.
A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure-Smith to prove an analogue of Deligne's conjecture for Loday algebras.
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 …
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.
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.
We show that if $(M,\tensor,I)$ is a monoidal model category then $\REnd_M(I)$ is a (weak) 2-monoid in $\sSet$. This applies in particular when $M$ is the category of $A$-bimodules over a simplicial monoid $A$: the derived endomorphisms of $A$ then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.
Generalised Joyal disks, $Θ_d$-colored $(d+1)$-operads, and higher Deligne conjecture
本文构造了颜色范畴为$\Theta_d$的染色$(d+1)$-operad $\mathbf{seq}_d$，推广了Tamarkin的2-operad，并证明其在拓扑和dg凝聚中可缩，为高维Deligne猜想提供关键工具。
In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace opera…
First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spac…
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…
A Deligne conjecture for prestacks
本文证明预叠的 Deligne 猜想类比，通过证实 Hawkins 关于 $\mathsf{Quilt}$ operad 正次同调消失的猜想，赋予 Gerstenhaber–Schack 复形自然的 $E_2$-代数结构。
The differential graded Verlinde Formula and the Deligne Conjecture
本文研究模范畴产生的微分分次模函子，证明环面映射类群中特定元素$S$的作用可将微分分次Verlinde代数转化为Deligne猜想提供的$E_2$结构，并在半单与非半单情形下分别归结为Verlinde公式及其非半单推广。
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevich's conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology complex of an n-algebra is an (n+1)-algebra. This generalizes a conjecture by Deligne for n=1, now proven by several authors.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。