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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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2 Answers 2

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

http://lanl.arxiv.org/abs/math/0507286

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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