Florin Felix Nichita and Calin Popescu: Entwined Bicomplexes p.161-176

Abstract:

We associate with each (co)algebra a bicomplex by encoding the
(co)multiplication in one of the differentials, and the (co)unit in the other. Akin to the (co)bar construction, the construction enables us to encode a morphism entwining an algebra with a coalgebra as a bicomplex morphism entwining the associated bicomplexes.

Next, we consider a related simplicial construction which associates a simplicial module with each (co)algebra. In this case, an entwining structure yields a simplicial map entwining the associated constructions, on one hand, and a chain map entwining the chain complexes on the constructions, on the other. Furthermore, the components of the simplicial entwining combine to yield another chain map between the chain complexes on the entwined products of constructions. The two chain maps thus obtained turn out to be compatible (they commute) up to chain homotopy with any pair of natural chain transformations for the Eilenberg-Zilber theorem which are the respective identities in dimension zero.

Key Words: (co)algebra, entwining structure, (co)bar construction, simplicial module, bicomlex.

2000 Mathematics Subject Classification: Primary: 16W30,
Secondary: 18G30, 18G35.

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