21

Can you, please, recommend a good text about algebraic operads?

I know the main one, namely, Loday, Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also there are notes by Vallette "Algebra+Homotopy=Operad", but they don't have much information and are too combinatorial. So what I am looking for is a pretty concise introduction to the theory of algebraic operads, that will be more algebraic then combinatorial, and that will give enough information to actually start working with operads.

Thank you very much for your help!

Edit: I have also found this interesting paper Modules and Morita Theorem for Operads by Kapranov--Manin. Maybe it's a bit too concise for the first time reading about operads, but it has a lot of really nice examples and theorems.

There are also notes by Vatne (only in PostScript).

evgeny
  • 1,990
Sasha Patotski
  • 1,266
  • 3
    Have you actually looked inside the book of Loday and Vallette? It is a large book, but if you know some stuff about associative algebras, many things can be skipped, and it is too well written to not be enjoyable. The "Leitfaden" included there (and reproduced on the webpage http://math.unice.fr/~brunov/Operads.html) would be very helpful to navigate. – Vladimir Dotsenko Aug 15 '13 at 11:38

5 Answers5

9

Benoit Fresse's book Modules over Operads and Functors is masterful.

Additionally, here are a couple of very good survey articles and notes from conferences:

AMS "What is..." article written by Stasheff

Expository article by Shenghao Sun

Notes from Algebra, Topology, and Fjords Conference

darij grinberg
  • 33,095
David White
  • 29,779
6

The book of Markl, Stasheff and Shnider is also a standard reference.

Also, a good jumping-in point could be Ginzburg and Kapranov's "Koszul duality for operads".

Dan Petersen
  • 39,284
  • 2
    I prefer the MSS book above the rest of references because it treats operads in many contexts. Other sources concentrate in linear operads, or some other specific kinds of operads. That book is not perfect, but it is the best available starting point. – Fernando Muro Aug 09 '13 at 18:51
  • 7
    Except of course that the actual definition of operads it gives is incorrect. A crucial equivariance formula is missing. The request is for something concise. There are two brief introductions to operads in general categories on http://www.math.uchicago.edu/~may/PAPERSMaster.html, numbers [84] and [85]. – Peter May Aug 10 '13 at 00:50
  • 2
    The MSS book is great, but some parts of it feel written hastily. Maybe it feels that way because in some 10 years since it was written, many things about operads were spelled out in a much cleaner way, like in Loday--Vallette. – Vladimir Dotsenko Aug 15 '13 at 11:35
2

Since both the following references appeared significantly later than the OP, it seems useful to add:

Peter Heinig
  • 6,001
2

The book by M. Bremner and V. Dotsenko titled Algebraic Operads: an algorithmic companion (published in 2016) is (in my perhaps biased opinion) a must-have for those wishing to complement their reading of Loday--Vallette. As the authors explain :

It is fairly accurate to say that the aim of this book is to create an accessible companion book to [180] which would, in the spirit of [64] contain enough hands-on methods for working with specific operads: making experiments, formulating conjectures and, hopefully, proving theorems, as well as, in the spirit of [252], include enough interesting examples to stimulate the reader toward those experiments, conjectures and theorems.

As the back-matter explains, it contains a systematic treatment of Groebner bases in several contexts, starting with non-commutative polynomials, and then moving to richer structure like twisted and shuffle algebras, and operads (ns, shuffle, symmetric), the main topic of the book. Like the book of Loday--Vallette, many instances of the book record relatively recent results concerning operads and related structures, and at the same time provides the reader with many challenging exercises (sometimes prompting them to use a CAS, if necessary) that provide invaluable insight for those aiming to make concrete computations using rewriting systems and their kin to study and prove results about operads.

[64] is Ideals, Varieties and Algorithms by Cox, Little and O'Shea, [180] is Algebraic Operads by Loday and Vallette and [252] is Combinatorial and Asymptotic Methods in Algebra by Ufnarovski.

Pedro
  • 1,534
0

A recent good book is Algebraic Operads by Jean-Louis Loday and Bruno Vallette.

evgeny
  • 1,990
Al-Amrani
  • 1,427
  • 1
    I've mentioned this book in my question. It really is a good book, but it's too big to read it as a quick introduction. But thank you for the link! – Sasha Patotski Aug 09 '13 at 18:21
  • Sorry, I did not notice that you know already Loday-Valette's book ! – Al-Amrani Aug 09 '13 at 18:22
  • 2
    Next September 4-6 (2013), I shall attend the meeting on "The Mathematical Legacy of Jean-Louis Loday" here in Strasbourg. In case I'll come across something interesting for you (as you ask), I'll signal it to you. – Al-Amrani Aug 09 '13 at 18:41
  • 5
    Just now, I attended Vladimir Dotsenko' talk : "Beyond quadratic operads". It was excellent; very clear, giving essential keys and links for understanding . He showed how Gröbner bases are a powerful tool to deal whith operads. See his homepage; he is a MathOverflow user.Ask him copies of his (general) talks on the subject (many meetings). – Al-Amrani Sep 05 '13 at 10:13