The Operads As and A

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


In this chapter, we first treat in detail the operad encoding the category of associative algebras along the lines of the preceding chapters. This is a particularly important example, because associative algebras are everywhere in mathematics, and because it will serve as a toy-model in the theory of operads. We prove that this operad is Koszul and we make explicit its Koszul dual and its Koszul resolution. In this way, we recover the notion of homotopy associative algebra introduced by Jim Stasheff. Finally, we give the Homotopy Transfer Theorem for this kind of algebras.


Chain Complex Associative Algebra Homology Class Algebra Structure Monoidal Category 
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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