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An Invitation to Applied Category Theory

Brendan Fong and David I. Spivak
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
David Jaz Myers
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An Invitation to Applied Category Theory is an invitation to category theory. Each of its seven chapters is a window on a different area of potential application -- database management, circuit diagrams, collaborative design, to name a few -- paired with an introduction to a categorical framework that can be used to reason about that area. Much like its role in mathematics, Category Theory is here applied as a framework, a way of reasoning about various applications of interest that is modular and transferrable (by being so general). In particular, you will find no grand theorems solving open problems in other fields; instead, you will find tools for thinking made real with examples from outside of pure mathematics. 
String diagrams give a poignant example of the applied categorical approach. A string diagram resembles a series of beads connected by strings. In this book string diagrams are used for many purposes. In Chapter 2, they appear as flow charts to describe composite processes (in a symmetric monoidal preorder). In Chapter 4, they describe the interlocking constraints that arise in the collaborative design of a project (working in the compact closed category of profunctors). In Chapter 5, they are used as signal flow graphs and then as matrices in a graphical approach to linear algebra. In Chapter 6, they form circuit diagrams (in the hypergraph category of decorated cospans). Over the course of these chapters, Fong and Spivak develop tools for working with string diagrams across these various applications, unifying what is common in them and laying the ground for analogies to pass between them. 
The book begins with an introduction to category theory, disguised as an introduction to the theory of orders. Since the proofs for categories are just more intricate versions of the proofs for orders, this nicely lays a ground of intuition for the remainder of the book to rest on. The book is not, however, a comprehensive introduction to category theory. For example, the Yoneda lemma is only given for orders, and without drawing on examples from pure mathematics, the notions of general limit and colimit (which are defined) are not fleshed out. Concepts are introduced as they are used, and are always written with the general theory that underlies them in mind.
An Invitation to Applied Category Theory is clearly and entertainingly written, and provides a great entry into the world of applied category theory. It is chock full of concrete examples and illustrated with clear diagrams. Exercises are interspersed throughout the text with full and comprehensive solutions in the back. The book is well suited for self-study; this reviewer mentored an undergraduate with a background in mathematical biology in reading a pre-print of the book over the course of a semester, and the student found the book engaging, learned the elements of category theory, and decided to continue studying from the book after the semester was over. In short, Fong and Spivak will whet your appetite for learning about categories and how they -- and the categorical way of thinking -- can be applied in and beyond mathematics. And they will give you the means to do that in a self-contained text.
There is an omission whose absence stands out to this reviewer: (co)homology, and in particular its use in topological data analysis. Category theory historically arose to organize and clarify methods in homological algebra in order to solve particular cohomological problems arising in topology; it continues to play a decisive role in (co)homology theories applied outside of pure mathematics (such as persistent homology). It's a shame to not have this topic treated by these authors as they did the other sketches, given their admirable way of weaving abstract categorical machinery and example applications.  
This book can be recommended to anyone with an interest in applied category theory and the ways it facilitates analogies and cross-pollination between applied fields.
David Jaz Myers is a Ph.D. student at Johns Hopkins University, studying category theory and homotopy type theory and wondering what it means to be a thing.