The essays in Categories for the Working Philosopher are an excellent illustration of why category theory might interest a “working” philosopher—that is, a philosopher whose day job is more or less independent of the foundations of mathematics. Each contribution describes some substantial application of categorical concepts to a specific philosophical issue with a minimum of technical overhead. This volume is an introduction to philosophical applications of category theory, not its mathematics, so it does not try to offer a systematic exposition of the main methods, definitions, and results of the theory. But if you want a sense of how category theory appears in contemporary philosophical research, you could scarcely do better than the essays in this collection.

Each of the 18 essays is self-contained, so you can dive in with whichever contribution is closest to your philosophical interests. But one of the strengths of this volume is the extensive and fruitful interaction among the different essays, and various natural groupings emerge from it. Elaine Landry gives one rough and ready classification in her editor’s preface: the first 11 chapters concern applications of category theory to the “pure” domains of logic and mathematics, and the remaining 7 “applied” chapters deal with topics in physics, biology, and formal philosophy of science. Alternatively, one might organize the contributions by the features of category theory they exploit. In the rest of this review I will attempt such an organization, offering one view on the main themes of this collection and how readers might make use of categorical concepts in their own work.

Category theory has undergone a number of radical shifts since its baptism 75 years ago, and it would be impossible to catalog here all of its core features over that period or even all of the features that appear in this volume. But roughly four of these features stand out: category theory’s distinctive abilities of abstraction, its treatment of dependence, the ease of formulating mathematically natural concepts in categorical terms, and the usefulness of category-theoretic constructions in other areas. Eilenberg and Mac Lane’s original development of category theory was motivated by problems of the following kind: given some set of instructions for gluing together rubber bands, find the number of holes in the resulting shape. A useful solution to this problem must track the complicated dependence of the answer on the gluing steps while recognizing that the final answer naturally abstracts away from irrelevant details like the steps’ specific order. And it must allow a natural translation of the flexible geometric description of the problem into a more rigid algebraic one, if we would like a computationally tractable result. Eilenberg and Mac Lane’s work relied heavily on a specific conception of the naturality at issue in these conditions, and they introduced category theory as a general axiomatic treatment of naturality in this sense—which, as they saw, appears throughout mathematics.

One group of essays engages with category theory’s particular flavor of abstraction. Colin McLarty argues that the category theory affords a formal theory of sets that hews more closely to the practice of most mathematicians than a highly structured formalization like Zermelo-Fraenkel set theory with choice (ZFC). Steve Awodey and Jean-Pierre Marquis each discuss the features of particular category-theoretic foundational frameworks that attempt to dispense entirely with the implementation details that distinguish models of the same mathematical objects—univalent foundations for Awodey and first-order logic with dependent sorts for Marquis. David I. Spivak sketches a view on which categories are mathematical models of mathematical models; in particular, category theory models those features of mathematical models that make a difference to their relations with other models. Finally, Kohei Kishida describes a framework that helps to tame the proliferation of semantics for quantified modal logic by abstracting away from particular implementations like theories of transworld identity or counterpart theories.

A second cluster of papers exploits category theory’s ability to finely discriminate between kinds of dependence. Samson Abramsky taxonomizes different quantum contextuality phenomena in which the global consistency of some data depends on more than their local consistency. James Owen Weatherall formalizes intuitive claims that one physical theory has more structure than another with a categorical characterization of the collection of models that coincide when you “forget” about the features of the former that depend on structures absent from the latter. Andrée Ehresmann’s contribution models the interdependence of levels of complexity in a biological or cognitive system, as well as their asymmetric time dependence. Hans Halvorson and Dimitris Tsementzis suggest that formal philosophy of science can profitably model a scientific theory with a category of models; for example, this perspective affords a more refined picture of a formal theory’s dependence on syntax. Landry’s own contribution argues that many attempts to use category theory in the service of structural realism often target the structure of the theory, rather than the world. However, she argues, category theory can be used to articulate a coherent concept of relations that do not depend on relata.

Category theory’s treatment of abstraction and dependence is valuable insofar as they allow you to neglect precisely the details you want to neglect while tracking dependencies that interest you: you generally do not want to choose Kuratowski’s model of ordered pairs or Quine’s, you just want a function that outputs ordered pairs to be the same thing as an ordered pair of functions. In many areas of mathematics the level of abstraction offered by category theory is exactly the level you naturally want. Combined with the in-principle possibility of formulating any mathematical notion in category-theoretic terms, this ability to identify mathematically natural concepts has led to claims that category-theoretic approaches to the foundations of mathematics are more natural and convenient than membership-based foundations like ZFC.

Two of the essays deal explicitly with this general claim that category theory offers a potentially superior approach to foundations. Michael Ernst gives a concise introduction to the debate over this claim so far and highlights a few questions that deserve further exploration. In a second contribution to this volume, Marquis argues that only a categorical framework can adequately account for the concept of a “canonical” map; he gives constraints on an account of canonicity and sketches a view on which canonicity is a central organizing principle in the architectonic of mathematics.

The rest of the chapters in this volume discuss applications of category theory to issues in other areas that philosophers do or ought to care about. These applications are enabled by category theory’s ability to model abstraction and dependence, and so most could be put in one or more of the above groups. I have put them in this last class because they are concerned with particular category-theoretic structures. For example, John L. Bell’s contribution provides a systematic introduction to functorial semantics—a common category-theoretic approach to model theory—as well as the concept of an elementary topos, which is the central object in the category-theoretic treatment of logic. Joachim Lambek defines a particular category that encodes the algebra of currents for a charged matter field in six-dimensional special relativity, which he uses to prove the conservation of probability density in that theory.

One pair of papers deals with the structures of higher-category theory. Michael Shulman’s contribution gives an accessible introduction to homotopy type theory and univalent foundations, which takes homotopy types (a.k.a. ∞-groupoids) as its basic objects, and explains some of the motivation for taking these objects as basic. David Corfield notes that philosophical engagement with geometry has essentially stalled in the geometry of the 1930s but argues that contemporary research in geometry has been made especially approachable to philosophers by the development of a framework in which homotopy types are basic.

The two remaining essays concern monoidal categories and their cousins, multicategories. A family of “coherence” results in basic category theory imply that these structures support a graphical calculus that amounts to pushing around decorated strings. J. R. B. Cockett and R. A. G. Seely use these categories to formulate a category-theoretic proof theory of the cut rule, which is formalized by composition in a category. Cockett and Seely give a modular treatment of structural rules and outline the resulting graphical calculus of rewriting rules. Bob Coecke and Aleks Kissinger use the monoidal structure of finite-dimensional Hilbert spaces to develop a graphical language for quantum mechanics that makes obvious some quantum phenomena like teleportation.

All told, the essays in this collection provide an accessible introduction to a great variety of ways philosophers can use category theory. They are almost uniformly high quality (which made it too hard to single out some for special attention in this review). They also provide broad coverage of the area: the only topic that felt missing was the application of category theory to formal semantics, and even that receives some mention. I would advise any working philosophers to read the contributions relevant to their interests.