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Twisting Morphisms

  • Jean-Louis Loday
  • Bruno Vallette
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 346)

Abstract

In this chapter, we introduce the bar construction and the cobar construction as follows. A twisting morphism is a linear map f:CA, from a dga coalgebra C to a dga algebra A, which satisfies the Maurer–Cartan equation:
$$\partial (f) + f\star f =0 .$$
The set of twisting morphisms Tw(C,A) is shown to be representable both in C and in A:
$$\operatorname{Hom}_{\mathsf{dga}\ \mathsf{alg}}(\Omega C, A) \cong \mathrm{Tw}(C,A) \cong \operatorname{Hom}_{\mathsf{dga}\ \mathsf{coalg}}(C, \mathrm{B}A).$$
Then we investigate the twisting morphisms which give rise to quasi-isomorphisms under the aforementioned identifications. We call them Koszul morphisms.

Keywords

Spectral Sequence Chain Complex Algebraic Topology Adjoint Functor Convolution Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jean-Louis Loday
    • 1
  • Bruno Vallette
    • 2
  1. 1.IRMA, CNRS et Université de StrasbourgStrasbourgFrance
  2. 2.Lab. de Mathématiques J.A. DieudonnéUniversité de Nice-Sophia AntipolisNiceFrance

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