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Is there a good introduction to

1.) Tensor (co)algebras on graded vector spaces ?

2.) Tensor (co)algebras on graded modules ?

In the research field of $L_\infty$-algebras there is some stuff, but nothing I know is a comprehensive introduction, including proofs.

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One pedagogical treatment of 1) is Chapter VIII from Marco Manetti's Lectures on deformations of complex manifolds

Another reference is Quillen's Rational homotopy theory, Appendix B. It is published in The Annals of Math., Vol.90, No.2, 1969.

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Thanks, on a short look on the first reference, it is indeed comprehensive. Nevertheless it says nothing about the situation on modules. (Like the tensor (co)-algebra on sections of a vector bundle) – Nevermind Feb 26 '12 at 10:51
  • A very nice book wit tons of examples is "Rational homotopy theory" by Felix, Halperin and Thomas. In particular, the section on graded differential algebras begins at pag. 40.

  • The book "Seminari di Geometria Algebrica" (1998-1999), Scuola Normale Superiore di Pisa (it is in English and probably available online) contains many details on tensor structures on dg vector spaces. I would start by having a look at the third chapter of the book, which is nothing but this paper by M. Grassi. Section 2 contains definitions and some results.

  • If you understand French, then the first chapter of Hasegawa's thesis contains a systematic analysis of differential graded and $A_\infty$-structures.

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