P. Hu, I. Kriz, A. A. Voronov
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.
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Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild cohomology and factorization (also called chiral) homology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum.
We give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristic is always a restricted Lie algebra.
Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras
本文通过范畴上同调与 incidence 代数的上同调研究图状 Hochschild 上同调，探讨了 Gerstenhaber-Schack 复合体与 Baues-Wirsching 猜想的联系，并计算了有限滤子 simplicial 复合物的图状 Hochschild 上同调。
We present a study of the homological algebra of bimodules over $A_\infty$-algebras endowed with an involution. Furthermore we introduce a derived description of Hochschild homology and cohomology for involutive $A_\infty$-algebras.
We give a method for computing factorization homology of $\oper{E}_n$-algebra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. We then show how to compute higher Hochschild homology and cohomology when the algebra is étale in a sense that we make precise. As an application, we compute higher Hochschild cohomology of the Lubin-Tate ring spectrum.
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.
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…
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these polynomials can be computed using Stasheff polyhedra and results from my previous paper GT/0207042.
Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gers…
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。
In this note, we define the notion of a cactus set, and show that its geometric realization is naturally an algebra over Voronov's cactus operad, which is equivalent to the framed 2-dimensional little disks operad $\mathcal{D}_2$. Using this, we show that the Hochschild cohomology of a Poincaré algebra A is an algebra over (the chain complexes of) $\mathcal{D}_2$.
An open-closed string analogue of Hochschild cohomology
本文证明开闭同伦代数可自然产生具有Gerstenhaber代数结构的开闭Hochschild上链复形，并发展开闭括号关系及扩展其微分的A-infinity结构。
Cyclic brace relation and BV structure on open-closed Hochschild cohomology
本文研究开闭同伦代数的Hochschild上同调，在循环且带单位的条件下，利用循环括号运算的上链级恒等式构造出BV代数结构。
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…
This paper investigates if a differential graded algebra can have more than one $A_\infty$-structure extending the given differential graded algebra structure. We give a sufficient condition for uniqueness of such an $A_\infty$-structure up to quasi-isomorphism using Hochschild cohomology. We then extend this condition to Sagave's notion of derived $A_\infty$-algebras after introducing a notion of Hochschild cohomology that applies to this.
Let A be an A_\infty ring spectrum. We use the description from [2] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A_\infty structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A_\infty structures on A. As an example, we study how topological Hochschild cohomology of Morava K-theory varies over the moduli space of A_\infty structures and show that in the…
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…
Hochschild cohomology and deformation theory of stable infinity-categories with t-structures
本文研究了配备左完备 t-结构的稳定可呈现无穷范畴的形变理论，证明其 Hochschild 上同调控制了该范畴关于左完备 t-结构的形变。
We consider an $A$-linear stable infinity-category $\mathcal{C}$ and the pair $(\mathcal{HH}^\bullet(\mathcal{C}/A),\mathcal{HH}_\bullet(\mathcal{C}/A))$ of the Hochschild cohomology spectrum (Hochschild cochain complex) and the Hochschild homology spectrum (Hochschild chain complex). The purpose of this paper is to provide a moduli-theoretic interpretation of the algebraic structure on the Hochschild pair of $\mathcal{C}$. The algebraic structure on the Hochschild pair is encoded by means of a two-colored topological operad called Kontsevich-Soibelman operad. The notions of cyclic deformation…
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 perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.
When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild (co)homology, presented by Pirashvili and Anderson, to study deformations seems only natural though the current definition allows only symmetric bi-module coefficients. In this paper we present an extended definition for higher order Hochschild cohomology which allows multi-module coefficients (when the simplicial sets $X_{\bullet}$ are accommodating) which…
$ L_\infty $-spectral sequences for Hochschild cohomology
本文提出经由 Maurer–Cartan 扭曲保留上同调更高 $L_\infty$ 结构的谱序列策略，并用于恢复穿孔曲面 wrapped Fukaya 范畴 Hochschild 上同调的形式性。
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