Twisting Morphisms

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


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.


Spectral Sequence Chain Complex Algebraic Topology Adjoint Functor Convolution Algebra 
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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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