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Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories

Hiro Lee Tanaka, Lukas Müller, Araminta Amabel, and Artem Kalmykov
Publication Date: 
Number of Pages: 
Springer Briefs in Mathematical Physics
[Reviewed by
Ittay Weiss
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Each of the three subjects in the title can serve as the title for a rather thick book. In fact, the latter two already do. Factorization homology, a homology theory for manifolds, is a relatively new and hot topic of research. While a single book devoted to it is not (yet) in existence, there is certainly a growing body of research articles, several of which written by Hiro Lee Tanaka, a leading expert and the deliverer of the lectures captured faithfully in the book under review.
The Springer Briefs series intends to deliver news from the frontiers of research. In this case, the material was captured by three attendees of the 2019 Summer School on Geometric Representation Theory and Low-Dimensional Topology held in Edinburgh, Scotland. The presentation appears to be, intentionally, very true to the delivery style in real-time. One can easily conjure up, between the lines, an image of the lecturer delivering the content to his audience.
So, how does one squeeze three huge topics, each well known to be fraught with great technical difficulty, into 82 pages? Well, luckily, one does not. Perhaps a good way to describe the book is that it is a pamphlet. This is not meant negatively at all. Factorization homology has a very slick definition in the language of \( \infty \)-categories, a rapidly growing subject with different (pretty much) equivalent approaches, all requiring significant effort to understand. Topological field theory has been around for some time and is equally challenging. The reporters tell us that “The aim of these lectures is to give an expository and informal introduction to the topics in the title. The target audience was imagined to be graduate students who are not homotopy theorists.” Consequently, not only is no assumption of prior knowledge of \( \infty \)-categories made, there is also no assumption that the reader has ever heard of \( \infty \)-categories, or that she has any interest in them.
Some detail on the content will aid in expressing the didactic approach.  Chapter 1 describes a miracle and some of its consequences. The miracle appears on page 5: “an algebraic operation turns out to be ”encoded” in something geometric and rather simple.” The algebraic operation is a unital associative algebra and the geometric encoding is a certain monoidal functor on the category of very simple objects, namely oriented 1-dimensional disks. The consequences discussed immediately upon witnessing the miracle lead to questions of higher-dimensions, a first look at \( \infty \)-categories, and the swift introduction of factorization homology. All of this is done within the first twenty pages by grounding the discussion around the easily illustratable miracle, without getting weighed down by the details of \( \infty \)-categories. Quite beautifully, the narration follows the inevitable path one is compelled to take once the miracle is internalized. If the reader is not hooked at that point, he should either re-read the book from page 1 or put it back on the shelf. If, instead, the reaction is “wow, this is really, really cool”, then chapter 2 will take the reader to the shores of \( \infty \)-category land.
I must confess that \( \infty \)-categories do not need to be ’sold’ to me. I am fully on board with the \( \infty \)-camp. Perhaps for that reason, I find the book’s take on the subject to be slightly (but really only slightly) weak. The opening paragraph of Chapter 2, the interlude on \( (\infty, 1) \)-categories, suggests that: “If the reader does not care for this chapter, they may soon find out, and they may just as soon skip to the next section.” Such an announcement may already seed the reader with antagonism toward a subject of central importance to the core material of the book.
Contrary to a possible expectation of ominous \( \infty \)-details to follow, the narration continues in the same friendly manner begun in chapter 1. Chapter 2 is very short, and so really one should not expect to see much technical detail. Instead, what one finds there is a good quick intro, pointers for further reading, and a sufficient introduction to nerves of categories to understand Joyal’s approach to \( \infty\)-categories via weak Kan complexes. In particular, the chapter delivers on the intention to “emphasize the importance of either not having a notion of composition inside a category, or of a functor not ”respecting” on the nose some notion of composition between a domain and codomain \( (\infty, 1)\)-category” declared in the opening paragraph.
Trusting that Chapter 2 provided sufficient motivation for the utility of \( \infty \)-categories and created a strong enough illusion that one can happily argue \( \infty \)-categorically without really having the details spelled out, Chapter 3 builds on the case of dimension 1 from Chapter 1 and goes on to consider factorization homology in higher dimensions. This includes a discussion of various classical structures, e.g., Hochschild chains and \( E_{n} \)-algebras, and again the presentation is highly geometric and the slick power of the \( \infty \)-categorical machinery is nicely put to work (after all, factorization homology is nothing but a Kan extension).
The final chapter continues in the same inviting fashion and defines the basics of cobordism theory. This time the definition is given in dimension n (with the mandatory pictures of some pairs of pants) and examined in detail in dimension 1. The reader with no prior knowledge of the subjects in the title of the book will, at this point, have an idea of topological field theory, an appreciation of the utility of \( \infty \)-categories, and the ability to articulate the definition of the new and exciting factorization homology theory. She will also have a useful list of sources for further reading at the end of each chapter and more than enough exercises if she wants to take it more seriously. A more experienced reader is likely to quite enjoy the ride that ties together various classical notions through one single Kan extension.
Ittay Weiss is senior lecturer in Mathematics at the University of Portsmouth, UK, interested in category theory, and in applications in topology, computer science and physics.