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Colored Operads

Donald Yau
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 170
[Reviewed by
Ittay Weiss
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Before the review proper I must address the color issue. The term “colored operad” is an unfortunate historical artifact originating from the fact that the first operads considered were all monochromatic, they all had just one color so there was no need to name this concept at all. But then certain variants of the original operads appeared and those were somewhat hard to digest. Those variants were in fact nothing but operads with two colors, and soon thereafter operads with any set of colors became part of the package. However, using “colors” to refer to the objects of an operad is simply an ad-hoc reaction to the need to distinguish among different things, and why not colorfully. But our understanding of operads has shifted significantly; we now know that an operad is simply the multi-input version of a category. The usage of the term “colored operad,” which was never prevalent, is now nearly extinct. It is unfortunate to see the title reiterating a temporary ill-chosen term! One must hope never to see a book titled Introduction to Colored Monoids instead of Introduction to Categories.

Operad theory is an increasingly important area of research with applications in mathematics, physics, and computer science. An introductory level book on the subject is very much welcomed, particularly that such texts are scarce; the only other introductory level text is Leinster’s Higher Operads; Higher Categories — a very different book in style and scope.

The book can be described as a gentle introduction to operads which boldly includes all of the technical details — with a strong emphasis on the words “all” and “boldly” — but avoids getting deep into the theory of operads. It can be likened to learning to swim in shallow water, paying full attention to each and every detail of the instruction. While we might certainly catch glimpses of the interesting objects that lie further ahead in the deep waters, they remain just beyond sight.

I will first address my reasons for this classification of the book, which is divided into four parts dealing with a graph formalism, an introduction to categories oriented toward operads, operads and their algebras, and free operads. The first part, the graph formalism, is particularly simple, easily accessible even to beginning undergraduate students. I find spending about 90 pages on it to be excessive, particularly considering the intended audience. This is reflected in the rather meager selection of exercises in part I — the material is too simple to allow for challenging exercises. Part II, the introduction to categories, concentrates on the topics that lead most naturally to operads. It involves most of the standard ingredients which are of a level of sophistication suitable for the target audience (i.e, adjunctions and Mac Lane’s coherence theorem) but not all (e.g., (co)limits). Part III introduces operads and their algebras, discussing a handful of examples, and then part IV is a study of free operads.

These choices of topics and their locations in the book are somewhat odd. I feel strongly that the category theory presented in part II really ought to be considered a prerequisite for reading the book (anybody seriously interested in operads should be expected to master a beginner’s knowledge of category theory). It would have been best placed as chapter 0. Part I, on the other hand, is highly technical, yet extremely intuitive and very simple — perhaps it should have been an appendix. Again, there is nothing stopping the reader from treating it as such. That would leave Part III and Part IV as the main topic of the book — a well written introduction to operads with much attention to the construction of free operads. Luckily, there is little to stop a reader from beginning with Part III.

What the book does quite well is provide a very well-written and highly detailed treatment of the basics of operads. Operad theory is notorious for classical introductory texts that are rather hard to follow, with many of the technicalities that this book puts in broad daylight only glossed upon (at best). There is certainly a need for a book which does everything slowly and carefully, for the sake of many a student finding the classical texts too much to handle. The book contains much valuable information and detail, which can potentially save a struggling newcomer into operad land many hours of frustration. In light of that, for the sake of an uninitiated wanderer facing the dilemma of having several sources to choose from, I would like to suggest a recommended way of reading the book: Treat one quarter as chapter 0, another one quarter as an appendix, with the remaining second half of the book a valuable, friendly, and well-written introduction to the basics of operads.

I will detail some dangers one may face if one follows the book in its intended order. First and foremost, the staggering amount of information that precedes the definition of operad may lead a reader to incorrectly infer that an operad is a terribly complicated thing to define; this is a dangerous misconception. Furthermore, Part I is, as already stated, very simple, but it is also very boring. The only motivation provided for traversing those 90 pages or so is that these concepts will be used later on, primarily in parts III and IV. That is certainly true, and that is why it should have been an appendix. Moreover, since the material really is very shallow a student reading it with ease may feel she is an operad savant. But understanding Part I with ease does not a talented operad theorist make. Part II, which should have been chapter 0, is a good introduction to category theory, but it is a bit of a missed opportunity since all there really is there is a geodesic to monoidal categories. The treatment in Mac Lane’s Categories for the Working Mathematician is, in my opinion, better.

So, if not as a first introduction to operads, how should one best use the book? I find the book highly useful as a companion to be consulted while going through any of the other texts on operads. When the going gets tough while one takes the first steps of reading a text where operads are used, one may find comfort in having a companion which will go through some of the details at great length.

There are however a few more warnings worth paying attention to. The rest of the review aims at providing further suggestions to the interested reader.

The graph formalism presented is, in my view, very unfortunate. First, it must be realized that the particularities of the formalism are irrelevant to the theory of operads, and further that it is not so much graphs that are of interest but rather trees. There are many formalisms for trees and not all are created equal. Different formalisms may suit different scenarios more than others. I find that point not to be sufficiently clearly emphasized to the reader. The literature on operads exhibits a plethora of tree formalisms, and it can be said with a fair amount of certainty that few experts care much about which formalism is used. That may be attributed to the fact that an ill-chosen formalism quickly becomes a burden rather than an asset when trying to use it to precisely get ideas across and validating theorems. The graph formalism chosen here is certainly among the common ones in the literature but, it is not optimally suited for operads, and I will explain why.

The graph formalism the author chose is the classical graph-theoretic one: a graph consists of a set of vertices and a set of edges where the edges are formed as pairs of vertices. In other words, the vertices are basic and the edges connect vertices. But there is a dual approach in which the edges are primitive and the vertices consist of information indicating which edges are put together to form the vertex. Most texts on operads (perhaps implicitly) take the latter approach, and for very good reasons.

Let us explain the advantages of the edge-centric formalism over the vertex-centric one. The need, in chapter one, to distinguish between what the author calls abstract vertices vs. vertices, is completely eliminated in the edge-centric formalism. Then, in chapter four (on page 45, particularly (4.2.2)), taking an edge-centric approach the operation of collapsing an internal edge is simplified, and in particular there is no need at all to identity vertices (since vertices do not exist as primitive objects). It should also be noted that the title “Associativity” for Section 4.4 is not quite in agreement with the content (the section should have been named “Commutativity”, or, better yet, it should have been named “Associativity and Commutativity” with the content suitably amended). Next, the grafting of trees in Chapter five is also greatly simplified in an edge-centric approach, particularly if one is content with grafting to be performed along outer edges and only to be defined along identical edges (in particular, no identifications are required). In sum, the graph formalism presented does not lend itself easily for the purposes of operad theory, and thus quickly becomes a burden rather than an illuminating and supporting conglomerate of ideas.

Part III starts off very well with a highly readable and friendly Chapter 10 presenting clear motivation for the definition of operad. Any reader who is even only partly familiar with categories can begin reading right there. Chapter 11 then delves into the definition of operad in a fairly standard fashion, followed by examining some variants and very elementary examples. Chapter 12 I find very odd, as it is, in light of the clear motivation and detail given in Chapter 10 and Chapter 11, simply the comment that since operads are the multi-input version of categories, categories are the special case of operads with only unary operations.

The reason operads exist is for their algebras, and these are presented in Chapter 13. Here I find the presentation to be suboptimal. The author follows directly in the footsteps of the first definitions of operads and their algebras, but that is not in line with the theme of the book, viewing operads as an extension of categories. It is much clearer to first consider morphisms of operads, and then define the algebra of an operad enriched in a monoidal category simply as a morphism from the former to the latter (after noticing that every monoidal category is canonically an operad, and emphasize the role of Mac Lane’s coherence theorem for this result to be true). Something along these lines is attempted in Section 13.9, but it falls short of truly illuminating the concept of an operad algebra simply as a morphism of operads with a very particular choice of codomain.


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Ittay Weiss is a Teaching Fellow at the University of Portsmouth, UK.

See the table of contents in pdf format.