On p. 130 of Algebraic Operads we come across the definition of the object in the book’s title; it’s pretty austere, viz. “a symmetric … operad is an S-module … endowed with morphisms … called composition map … the unit map, which make [the monad] into a monoid.” Happily this gives us a point of contact with something reasonably familiar, namely monoids (closure, associativity, and identity, but not necessarily inverses), even as it tosses something more esoteric at us in the form of S-modules. Well, on p. 122 we encounter these beasties: S-modules are families of right K[Sn]-modules indexed on the natural numbers, where K is the ground field (in the context of vector spaces) and, happily, Sn is the usual symmetric group: we’re dealing with old-fashioned group rings.
So the players on stage emerge from the shadows with at least some familiar features discernible. Nonetheless, all this is asking a lot: here we are already in Chapter 5 of the book, and only now do these things get defined — and the definitions are pretty fruity. I like commutative diagrams a lot (they proliferate like rabbits right after p. 130), but what’s the context, where are the connections, what’s the motivation?
Well, that’s what the book’s first four chapters are all about. It starts off with “Algebras, Coalgebras, Homology,” where Lie theory makes a necessary appearance. Thereafter it’s really Koszul theory on center stage with “Twisting Morphisms,” “Koszul Duality for Associative Algebras,” and “Methods to Prove Koszulity of an Algebra” (and the next time I play hangman, I’ll make it a point to work in “koszulity”!).
One defines “a quadratic data” [sic; datum?] to be a graded vector space (presumably over K) together with a graded subspace of the tensor product of that space with itself, together with “quadratic morphisms” (cf. p. 62). The focus falls on the quadratic algebra obtained by taking the quotient of the free associative algebra over the given vector space by the two-sided ideal generated by the chosen subspace of the space’s tensor square. And then (p. 70) a quadratic algebra, or more generally a quadratic data, is Koszul iff its Koszul complex is acyclic (i.e. has vanishing (co)homology). Finally, the Koszul complex construction is given on p. 68: it’s a marvelously algebraic bit of manouevreing, with the differential coming directly from a pretty natural twisting morphism; seeing that we’re lowering indices it’s really homology, and not cohomology. So there.
By the way, this all-important fifth chapter is introduced by the following quote by J. P. May: “The name ‘operad’ is a word that I coined myself, spending a week thinking about nothing else.” Given that May is still going strong at the University of Chicago, we’re dealing with some relatively young and vigorous stuff: modern algebraic topology without any apologies (as if it needs any!).
The upshot is that Algebraic Operads is very serious modern mathematics in action, with a non-trivial apprenticeship called for on the part of the reader, especially vis à vis algebraic topology. The authors note in their Preface, for instance, that “[o]ne of the main fruitful problems in the study of a given type of algebras is its relationship with algebraic homotopy theory,” and it is this context that provides the raison d’être for what they proceed to do in the pages that follow. To wit,
starting with a chain complex equipped with some compatible algebraic structure, can this structure be transferred to any homotopy equivalent chain complex? In general, the answer is negative. However, one can prove the existence of higher operations [!] on the homotopy equivalent chain complex, which endow it with a richer algebraic structure … It is exactly the problem of governing these higher structures that prompted the introduction of the notion of operad.
And one then comes across such luminaries of the game as Stasheff (already in the 1960s), Priddy (regarding Koszul duality, in the 1970s), Ginzburg and Kapranov (regarding associative algebras) and, in the words of Loday and Vellette themselves, “Getzler and Jones in the 1990s (part of the renaissance period).” The authors note that “[t]he application of Koszul duality theory for operads to homotopical algebra is a far-reaching generalization of the ideas of Dan Quillen and Dennis Sullivan.” There’s poignancy in this phrase in that Quillen passed away only last year and the current (November 2012) Notices of the AMS contains a long obituary section devoted to him, including a tribute by Jean-Louis Loday, one of the authors of the represent book, on pp.1402–1403. This poignancy is amplified by the dedication phrase succeeding the book’s title page: “À la mémoire de JLL”: Loday died in 2012. May God rest their souls.
This book is indeed a fine example of cutting edge algebraic topology. While the point of view centering on operads is some forty years old, major new trends emerged in the wake of the “renaissance” of the 1990s already alluded to above. This change in focus engendered “a shift from topology to algebra” culminating around the turn of the century with a work by Markl, Shnider, and Stasheff titled, Operads in Algebra, Topology, and Physics. Subsequently, i.e. in the present decade, “most of the basic aspects… have been settled and it seems to be the right time to provide a comprehensive account of algebraic operad theory. This is the purpose of this book.”
Little more remains to be said. Algebraic Operads is a fine introduction to a modern mathematical subject that is obviously both fascinating and fecund, and it is right that this book appears in Springer’s vaunted Grundlehren der mathematischen Wissenschaften series (or as it used to be called, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen — ah, I miss the old days …).
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.