Category theory and philosophy

The draft of a paper by Barry Mazur on what category theory tells us about identity is available: When is one thing equal to some other thing. Philosophy's encounter with category theory has not been its happiest. The trouble is that, as Mazur puts it, category theory shines its spotlight in a different direction to the traditional foundational languages. Looking through its lens it leads us to ask different questions about mathematics. The closest one will come to a direct clash is over identity or sameness. But many philosophers find it hard to understand category theory's position on this as first they want to know what 'stuff' goes to make up the things we are wondering are the same or not. Besides Mazur's piece, there's a very good account of the category theoretic position by Steve Awodey. Awodey is replying to the philosopher Geoffrey Hellman, so if you are coming from an Anglophone philosophical starting point, this should be especially helpful.

A recently establish publisher, Polimetrica, is bringing out a philosophy book at the start of next year - What is Category Theory? - including a contribution by myself. On their catalogue you will also find Generic figures and their glueings - A constructive approach to functor categories by Marie La Palme Reyes, Gonzalo E. Reyes, Houman Zolfaghari:

This book is a "missing link" between the elementary textbook of Lawvere and Schanuel "Conceptual Mathematics" and the much more advanced textbooks such as the one by MacLane and Moerdijk "Sheaves in Geometry and Logic."

Avoiding the complicated, fully fledged notion of a Grothendieck topos, whose very formulation presupposes a good deal of mathematical experience, this book introduces topos theory through presheaf toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the "generic figures". Several phenomena which distinguish toposes from the ordinary category of sets appear already at this simpler level.

Six easy to understand examples accompany the reader through the whole book, illuminating new material, interpreting general results and suggesting new theorems.

This book is aimed (via appropiate examples) at a beginner mathematician or scientist or philosopher who would like to take advantage of the richness of presheaf toposes to prepare himself or herself either for further study or applications of the theory described.

Of course, after categories come bi-, tri-, and tetra-categories, right up to omega-categories. There are even Z-categories, where Z stands for the integers. (The spectra of algebraic topology are examples of Z-groupoids.) The point is that sameness becomes a more subtle question as we climb the ladder. The most noticeable manifestation of this is where one says two categories are the same if they are equivalent. This means that there are a pair of functors going in opposite directions between them, such that their composites are not equal to the identity functor, but that there is a natural equivalence between them. This idea shows up in many fields by the name Morita equivalence (see, e.g., A bicategorical approach to Morita equivalence for von Neumann algebras). It's a little like saying that London and Oxford are more alike than London and Paris, because I can walk to Oxford from London and then retrace my steps, and while this isn't the same as resting in London, there's a continous set of paths mediating between the null journey and the Oxford round trip.

You can read about this in chapter 10 of my book and in this paper, in Yuri Manin's Georg Cantor and his heritage, and in 101 places on John Baez' site.