Koszul duality for operads

Ginzburg, Victor; Kapranov, Mikhail

doi:10.1215/s0012-7094-94-07608-4

Cited by 631 publications

(960 citation statements)

References 42 publications

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“…For a quadratic operad O we will denote by O _ the Koszul dual cooperad [7], [13]. Bar.O/ will denote the bar construction of the (augmented) operad O, and Cobar.C/ will denote the cobar construction of the (coaugmented) cooperad C. For an operad O (resp.…”

Section: Notationmentioning

confidence: 99%

The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal

Dolgushev¹,

Tamarkin²,

Tsygan³

2007

J. Noncommut. Geom.

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Abstract. The solution of Deligne's conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C .A/ of a regular commutative algebra A over a field K of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand-Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold. (2000). 53D55. Mathematics Subject Classification

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Section: Notationmentioning

confidence: 99%

The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal

Dolgushev¹,

Tamarkin²,

Tsygan³

2007

J. Noncommut. Geom.

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“…On our side of the duality, tree maps are described in terms of set maps between the sets of edges. The category of generic injections is roughly the opposite of the category of trees studied 1 by Ginzburg and Kapranov in their seminal paper [7]; the difference is that they exclude all trees with nullary nodes. In fact, most of the time they also exclude trees with unary nodes, and call the remaining trees reduced.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Functors and Trees

Kock

2010

International Mathematics Research Notices

152

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We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Ω of Moerdijk and Weiss; its main properties are described in terms of some factorisation systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

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“…In the differential graded setting over a field of characteristic 0, homology with trivial coefficients was defined by Ginzburg and Kapranov in [9] and by Getzler and Jones in [8]. Homology with coefficients was defined by Balavoine in [2].…”

Section: Gamma-homology Of P-algebrasmentioning

confidence: 99%

“…It is a cooperad. The Koszul construction with coefficients P ı KP ı P is equipped with a differential, such that .P ı KP ı P; @/ is quasi-isomorphic to P. For more details, we refer the reader to the initial article of Ginzburg and Kapranov [9] or the article of Fresse [6], of which we adopt the convention.…”

Section: Definition Of the Complexmentioning

confidence: 99%

Γ–homology of algebras over an operad

Hoffbeck

2010

Algebr. Geom. Topol.

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-homology of algebras over an operad ERIC HOFFBECKThe purpose of this paper is to study generalizations of Gamma-homology in the context of operads. Good homology theories are associated to operads under appropriate cofibrancy hypotheses, but this requirement is not satisfied by usual operads outside the characteristic zero context. In that case, the idea is to pick a cofibrant replacement Q of the given operad P. We can apply to P-algebras the homology theory associated to Q in order to define a suitable homology theory on the category of P-algebras. We make explicit a small complex to compute this homology when the operad P is binary and Koszul. In the case of the commutative operad P D Com, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras. 16E40; 18D50, 18G55, 18G60The classical homology theories of commutative algebras (Harrison homology in the differential graded setting over a field of characteristic 0, see Harrison [11], and André-Quillen homology in the simplicial setting over a ring of any characteristic, see Quillen [16] and André [1]) can be considered as homology theories associated to the commutative operad Com. There is another homology theory for commutative algebras, -homology (Gamma-homology in plain words, also called topological André-Quillen), which has been introduced by Robinson and Whitehouse in [18], and by Basterra in [3] (with a different point of view), to solve obstruction problems in homotopy theory. In the setting of [18], Gamma-homology is defined as the homology theory associated to an E 1 -operad (a cofibrant replacement of Com). This new homology can be defined in the context of differential graded or simplicial context or in the context of spectra, and gives the same result in each case (see Mandell [15]), in contrast with the usual André-Quillen homology.The purpose of this paper is to study generalizations of -homology in the context of operads.

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Koszul duality for operads

Cited by 631 publications

References 42 publications

The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal

The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal

Polynomial Functors and Trees

Γ–homology of algebras over an operad

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