A categorical approach to cyclic duality

Böhm, Gabriella; Ştefan, Dragoş

doi:10.4171/jncg/98

Cited by 25 publications

(6 citation statements)

References 19 publications

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“…for m P M . As mentioned in [BöS ¸t1,§A.7] and similarly as for comodules, a right Ucontramodule additionally induces a left A-action given by…”

Section: Contramodules Over Bialgebroidsmentioning

confidence: 95%

When Ext is a Batalin–Vilkovisky algebra

Kowalzig

2018

J. Noncommut. Geom.

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We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg-Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf-Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. As an application, we prove that the Hochschild cohomology of an associative algebra A is Batalin-Vilkovisky if A itself is a contramodule over its enveloping algebra A b A op . This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.

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“…for m P M . As mentioned in [BöS ¸t1,§A.7] and similarly as for comodules, a right Ucontramodule additionally induces a left A-action given by…”

Section: Contramodules Over Bialgebroidsmentioning

confidence: 95%

When Ext is a Batalin–Vilkovisky algebra

Kowalzig

2018

J. Noncommut. Geom.

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“…Using the adjunctions for M, V, W ∈ H M: We want to understand the center Z H M ( H M op ). First of all let us recall the notion of contramodules over bialgebroids (as it was defined in [2], generalizing the classical contramodules for coalgebras [8]).…”

Section: Anti-yetter-drinfeld Contramodules For Hopf Algebroidsmentioning

confidence: 99%

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

Kobyzev

Shapiro

2018

Appl Categor Struct

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We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.

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“…This object is precisely the one constructed in [3] as recalled in Section 2.4 above. This construction was generalized slightly in [5] to include right λ-coalgebra structures on arbitrary functors Y → X, rather than just objects of X; in this case y becomes a functor Y → H 0 (A, X) and the composite…”

Section: 4mentioning

confidence: 99%

“…Just as simplicial structure can be used to define homology, cyclic (or duplicial or paracyclic) structure can be used to define cyclic homology. In a series of papers [3,4,5], Böhm and Ştefan looked at what further structure than a comonad is needed to equip the induced simplicial object with duplicial structure; the main extra ingredient turned out to be a second comonad with a distributive law [2] between the two. They also showed that their machinery could be used to construct the cyclic homology of bialgebroids.…”

Section: Introductionmentioning

confidence: 99%

Hochschild homology, lax codescent, and duplicial structure

2018

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We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Böhm and Ştefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.

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A categorical approach to cyclic duality

Cited by 25 publications

References 19 publications

When Ext is a Batalin–Vilkovisky algebra

When Ext is a Batalin–Vilkovisky algebra

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

Hochschild homology, lax codescent, and duplicial structure

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