In fact I am only going to really comment (and that only briefly) on one of these four papers. For two of them relate to quantum mechanics; and to my great regret I quite lost my grip on such matters many years back. Here, Samson Abransky writes on ‘Contextually: On the borders of paradox’; and Bob Coecke and Aleks Kissinger contribute ‘Categorical quantum mechanics I: causal quantum processes’. Both papers can be downloaded from the arXiv and you can chase them up. And if you want to know more about Bob Coecke and Aleks Kissinger’s take on quantum mechanics, they have a very large book *Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning* (CUP, 2017) whose opening chapters are pretty accessible.

Back in the Landry collection, another paper is an eight-page note by the late Joachim Lambek on a ‘Six-dimensional Lorentz category’ (again the piece is downloadable). This, however, seems quite out of place in this volume. And indeed the author himself concludes ‘The two extra dimensions of time had been introduced for the sake of mathematical elegance and I have not settled on their physical meaning. For a while I had hoped that they might help to incorporate the direction of the spin axis, but did not succeed to make this idea work’ — hardly a ringing endorsement of the project. Best forgotten as far as I can see.

Which leaves James Owen Weatherall’s ‘Category theory and the foundations of classical space-time theories’: again, a version of this paper is downloadable.

I don’t know quite what Elaine Landry asked of her contributors. In her preface, however, she writes that ‘this book aims to bring the concepts of category theory to philosophers working in [a variety of] areas …

Moreover, it aims to do this in a way that is accessible to a general audience.’ And Weatherall’s piece is indeed clear and engaging. But does he actually show categorial ideas doing essential work?

His topic is various classical field theories which have, in an intuitive sense, “excess content” (they are, as it is said, gauge theories), and the aim is to use categorial ideas to analyse this notion of excess content. Without going into details here, the discussion is interesting and persuasive about the differences between various gauge theories. He sums up:

I have reviewed several cases in which representing a scientific theory as a category of models is useful for understanding the structure associated with a theory. In the context of classical space–time structure, the category theoretic machinery merely recovers relationships that have long been appreciated by philosophers of physics; these cases are perhaps best understood as litmus tests for the notion of “structure” described here. In the other cases, the new machinery appears to do useful work. It helps crystalize the sense in which [versions of classical Newtonian gravity and of electromagnetic theory] have excess structure, in a way that clarifies an important distinction between these theories and other kinds of gauge theories, such as Yang–Mills theory and general relativity. It also clarifies the relationship between various formulations of physical theories that have been of interest to philosophers because of their alleged parsimony. These results seem to reflect real progress in our understanding of these theories — progress that apparently required the basic category theory used here.

But the last claim does seem to overshoot. The basic category theory in question is just the invocation of the notion of a functor as a map between different models and their automorphisms, plus the idea that different functors can preserve different amounts of information, a general idea which is entirely available to someone who has met no category theory at all. In fact, Weatherall himself admits as much at the beginning of his paper:

Although some of the results I describe in the body of the chapter are non-trivial, the category theory I use is elementary and, arguably, appears only superficially.

He does give a promissory note that there are cases in the same neck of the woods to the ones which he discusses ‘where category theory plays a much deeper role’. But as things stand in this paper itself, the category theory indeed seems inessential.