Multivariable cochain operations and little 𝑛-cubes

McClure, James E.; Smith, Jeffrey H.

doi:10.1090/s0894-0347-03-00419-3

Cited by 97 publications

(166 citation statements)

References 18 publications

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“…This operad is also introduced (with different sign conventions) by J. McClure and J. Smith in their work on the Deligne conjecture (cf. [22,23]). To be precise, the Barratt-Eccles operad E operates on cochain complexes of simplicial sets and on Hochschild complexes of associative algebras through a morphism T R: E →X which we call the table reduction morphism.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial operad actions on cochains

Berger¹,

Fresse²

2004

Math. Proc. Camb. Phil. Soc.

140

306

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A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.Comment: 46 pages. Final versio

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Section: Introductionmentioning

confidence: 99%

Combinatorial operad actions on cochains

Berger¹,

Fresse²

2004

Math. Proc. Camb. Phil. Soc.

140

306

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“…The first part of this is immediate. For the second we construct the stabilization via a colimit whose maps actually add topology to the pieces p and explicitly give the E ∞ operations as the geometrization of the operations of [9]. If A is not semi-simple the action does not pass through the stabilization.…”

Section: Some Details On the Action Relevant To The Further Discussionmentioning

confidence: 99%

Noncommutative aspects of open/closed strings via foliations

Kaufmann

2008

Reports on Mathematical Physics

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We give a brief summary of algebraic aspects of string theory arising in the noncommutative geometry setting of foliations called string diagrammatics which we introduced jointly with Bob Penner. We furthermore discuss how this gives rise to actions on the Hochschild complex of a Frobenius algebra. We then explain how this leads to new quantum chains for loop spaces and a stabilization in the semi--simple case.Comment: 12 pages 3 figures. ROMP to appea

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“…Proof. (i) ⇒ (ii): In [Rickard 1991], it is shown that and admit a standard derived equivalence. Thus there is a bounded complex P of --bimodules which in each degree is finitely generated projective over and over .…”

Section: (I)mentioning

confidence: 98%

“…The classical Morita theory for derived categories [Rickard 1989;1991] can be extended to complexes of injective modules as follows.…”

Section: A Recollement For K(inj Kg)mentioning

confidence: 99%

Untitled

2008

ANT

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Let G be a finite group and k be a field of characteristic p. We investigate the homotopy category K(InjkG) of the category C(InjkG) of complexes of injective (= projective) kG-modules. If G is a p-group, this category is equivalent to the derived category D dg (C * (BG; k)) of the cochains on the classifying space; if G is not a p-group, it has better properties than this derived category. The ordinary tensor product in K(InjkG) with diagonal G-action corresponds to the E ∞ tensor product on D dg (C * (BG; k)). We show that K(InjkG) can be regarded as a slight enlargement of the stable module category StModkG. It has better formal properties inasmuch as the ordinary cohomology ring H * (G, k) is better behaved than the Tate cohomology ringĤ * (G, k). It is also better than the derived category D(ModkG), because the compact objects in K(InjkG) form a copy of the bounded derived category D b (modkG), whereas the compact objects in D(ModkG) consist of just the perfect complexes. Finally, we develop the theory of support varieties and homotopy colimits in K(InjkG).

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Multivariable cochain operations and little 𝑛-cubes

Cited by 97 publications

References 18 publications

Combinatorial operad actions on cochains

Combinatorial operad actions on cochains

Noncommutative aspects of open/closed strings via foliations

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