Algebraic operads. (English) Zbl 1260.18001

Loday, Jean-Louis; Vallette, Bruno

Grundlehren der Mathematischen Wissenschaften 346. Berlin: Springer (ISBN 978-3-642-30361-6/hbk; 978-3-642-30362-3/ebook). xxiv, 634 p. (2012).

An operad is an algebraic device encoding a type of algebras. Operad theory emerged as an efficient tool in algebraic topology in the 1960s in the work of Frank Adams, Peter May, André Joyal, Saunders MacLane and other topologisits and category theorists. In the 1990s, there occurred a renaissance of the theory in the development of deformation theory and quantum field theory, shifting from topology to algebra in the work of Victor Ginzburg, John Jones, Mikhail Kapranov, Yuri I. Manin, Ezra Getzler, Maxim Kontsevich and many others, which resulted in the first monograph [Zbl 1017.18001] a decade later. Now that two decades have passed since the above renaissance of operad theory, it seems the right time for a comprehensive account of algebraic operad theory, which is the aim of the present monograph. The monograph has three purposes, namely, firstly to provide an introduction to algebraic operads, secondly to give a conceptual treatment of Koszul duality, and thirdly to give applications to homotopical algebra.
The monograph consists of 13 chapters together with three appendices. The first four chapters (Chapters 1-4) deal with Koszul duality of associative algebras. The distinguishing feature of the authors’ treatment is to keep algebras and coalgebras on the same footing. Classical references on Koszul duality of associative algebras are [Zbl 0261.18016; Zbl 0625.58040; Zbl 0724.17006; Zbl 0962.13009; Zbl 1145.16009] and so on.
The next four chapters (Chapters 5-8) are concerned with algebraic operads and their Koszul duality. Chapter 6, which treats operadic homological algebra, is on the lines of [Zbl 0855.18006; Zbl 0855.18007; Zbl 1077.18007], while Chapter 8, presenting various methods to prove that algebras are indeed Koszul, is on the lines of [Zbl 1207.18009; Zbl 1208.18007; Zbl 0853.18005; Zbl 1140.18006; Zbl 1159.18001; Zbl 0855.18006; Zbl 0855.18007].
The succeeding four chapters (Chapters 9-12) are concerned with homotopy theory of algebras over an operad. Chapter 9 treats the operad encoding the category of associative algebras along the lines of the preceeding chapters. The purpose of Chapter 10 is to show that if the chain complex contains a smaller chain complex which is a deformation retract, then there is a finer algebraic structure on this small complex, the small complex with this new algebraic structure being homotopy equivalent to the original data. Chapter 11 follows essentially [Zbl 1245.55007; Zbl 0906.55005; Zbl 1218.18007; Zbl 1276.18009; Zbl 1151.53076]. Chapter 12 introduces the André-Quillen cohomology and homology of algebras over an operad after [Zbl 1283.18007; Zbl 1218.18007; Zbl 0999.18009; Zbl 0967.18004; Zbl 0883.17004; Zbl 0894.18008; Zbl 0855.18006; Zbl 0855.18007].
The last chapter (Chapter 13) presents several examples of operads besides the operad \(Ass\) encoding associative algebras already studied in Chapter 9. The three appendices are devoted to symmetric groups, categories and trees.

Reviewer: Hirokazu Nishimura (Tsukuba)

MSC:

18-02	Research exposition (monographs, survey articles) pertaining to category theory
18D50	Operads (MSC2010)
18G50	Nonabelian homological algebra (category-theoretic aspects)
55P48	Loop space machines and operads in algebraic topology
57T30	Bar and cobar constructions
16S37	Quadratic and Koszul algebras
16E99	Homological methods in associative algebras
18G55	Nonabelian homotopical algebra (MSC2010)
55U35	Abstract and axiomatic homotopy theory in algebraic topology
17A32	Leibniz algebras
17A30	Nonassociative algebras satisfying other identities

Keywords:

operad; Koszul duality; operadic homological algebra; homotopy operadic algebra; algebraic operad theory; homotopical algebra

Citations:

Zbl 1017.18001; Zbl 0261.18016; Zbl 0625.58040; Zbl 0724.17006; Zbl 0962.13009; Zbl 1145.16009; Zbl 0855.18006; Zbl 0855.18007; Zbl 1077.18007; Zbl 1207.18009; Zbl 1208.18007; Zbl 0853.18005; Zbl 1140.18006; Zbl 1159.18001; Zbl 1245.55007; Zbl 0906.55005; Zbl 1218.18007; Zbl 1151.53076; Zbl 0999.18009; Zbl 0967.18004; Zbl 0883.17004; Zbl 0894.18008; Zbl 1276.18009; Zbl 1283.18007