### Categories for the Working Philosopher

#### Posted by John Baez

Elaine Landry, in the philosophy department at U. C. Davis, is putting together a book called *Categories for the Working Philosopher*.

Here are the contributors and their (perhaps tentative) topics:

- Samson Abramsky — Computer Science, etc.
- John Baez — Applied Mathematics
- John Bell — Logic/Model Theory
- Bob Coecke — Quantum Mechanics and Ontology
- Robin Cockett — Proof Theory/Linear Logic
- David Corfield — Geometry
- Andrée Ehresmann — Biology
- Hans Halvorson — The Structure of Physical Theories
- Kohei Kishida — Modal Logic
- Jim Lambek — Special Relativity
- Jean-Pierre Marquis — First-Order Logic with Dependent Sorts
- Colin McLarty — Set Theory
- Michael Moortgat — Linguistics and Computational Semantics
- Michael Shulman — Univalent Foundations
- David Spivak — Mathematical Modeling
- James Weatherall — Spacetime Theories

It looks like a nice lineup!

I say the topics are tentative because I just changed my own. I’d been wanting to write about category theory and the foundations of applied mathematics, since I’d talked about that a while ago at U. C. Irvine. However, this seems like a bad idea for a couple of reasons.

First, I feel my thoughts on this subject aren’t best expressed at a philosophical level right now. I think I should first go ahead and *do* applied mathematics using category theory. And indeed, that’s one of the main things I’m doing these days, with the help of 7 grad students.

Second, a lot of authors in this book will be talking about more traditional regions of overlap between philosophy and category theory — like logic, and the foundations of mathematics and physics. Since I’m still interested in such topics, it makes sense to write something on this theme.

Third, since Mike Shulman will be writing about univalent foundations and Jean–Pierre Marquis will be talking about first-order logic with dependent sorts (or ‘FOLDS’ for short), I think it would be good to write about some general issues which both these initiatives address.

So, here’s a draft of an abstract of my paper.

### Notions of Sameness

The role of equality and other notions of ‘sameness’ in mathematics is fundamental, yet still far from settled. When we say $2 + 3 = 5$, a naive schoolchild may wonder why we are claiming different expressions are the same. When we say $5 = 5$, we have identical expressions on both sides of the equation, so this problem does not arise. Yet equations of this form, $x = x$, are of almost no use in mathematics! Ironically, the more different the two sides of an equation look, the more valuable it is.

A standard way of dealing with this is with the idea of ‘reference’. A formal theory consists of syntactic expressions and rules for manipulating them. A ‘model’ for such a theory, formulated in some meta-theory, maps expressions to mathematical entities in the meta-theory. In particular, for the theory of arithmetic, expressions such as 2 + 3 and 5 are mapped to natural numbers: we say they ‘refer’ to these numbers. The sentence ‘2 + 3 = 5’ is mapped to the truth value ‘true’ because 2 + 3 and 5 are mapped to the same natural number.

This is fine as far as it goes. However, the recourse to a meta-theory, and the explanation of equality in the theory in terms of equality in the meta-theory, suggests that we may be merely postponing the need to think about what equality really means.

The standard axioms obeyed by equality (reflexivity, symmetry and transitivity) certainly agree with our intuitions about the meaning of this concept. Yet these axioms are really the definition of an ‘equivalence relation’, and there are many other equivalence relations beside equality. This suggests that these axioms are capturing intuitions about what it means for two things to be ‘the same in a way’, not necessarily equal.

Leibniz’s principle of ‘identity of indiscernibles’ was a great step forward in understanding equality. He said that $x = y$ if and only if for all predicates $P$, $P(x) \iff P(y)$. The intuition is a powerful one: things with the same properties are equal. However, when mathematics was formalized on the basis of set theory in the early 20th century, this principle played little role. The reason is that it uses second-order logic — we are quantifying over the predicate $P$ — while for important technical reasons, these systems are usually formulated in first-order logic.

With the introduction of category theory, a series of important breakthroughs began, which gradually transformed the whole discussion of equality. It is time for philosophers to pay attention to this changed landscape.

First, category theory formalizes the concept of ‘isomorphism’. Roughly speaking, two things are isomorphic if they are the same in a way. An isomorphism is a specific way. There can be many such ways.

For example, a cube has 48 symmetries. Each of these is an isomorphism between the cube and itself. Since the cube is equal to itself, it follows that the cube is isomorphic to itself; this is nothing special, since it is true of any shape. But the fact that the cube is isomorphic to itself in 48 ways is something interesting.

In general, the isomorphisms from an object in a category to itself form a group. Thus, group theory is automatically brought into our discussion of ‘sameness’ when we formalize this concept using isomorphisms in a category. This is a powerful step, for many reasons. One is that group theory is a rich and lively branch of mathematics. Another is that ever since the work of Klein, we have been able to think of group theory as a distilled, concentrated approach to geometry. The example of the cube hints at this, but Klein and subsequent mathematicians went much further.

In addition to isomorphisms between an object and itself, we can also consider isomorphisms between objects that are not equal. This gives a ‘groupoid’: a powerful generalization of the concept of group. And groupoids naturally bring topology into the game. The reason is that for any space there is a groupoid with the points of that space as objects, and certain equivalence classes of paths in that space as isomorphisms.

What does this mean? It may sound technical, but it is fundamental. It means simply that two points are ‘the same in a way’ if we can get from one to the other by following some path. Furthermore, the path is the way. It is no coincidence that ‘way’ is sometimes used to mean ‘path’: we are capturing some ancient intuitions here!

In short, by formalizing the concept of sameness using isomorphisms instead of equality, we bring group theory, geometry and topology closer to the foundations of mathematics, giving them a kind of inevitability that they might not otherwise seem to possess. We also see that these three subjects are tightly connected. A lot of important mathematics, and also theoretical physics, flows from this realization.

For example, all the forces we understand in nature—electromagnetism, the strong and weak nuclear forces, and gravity—can be described as ‘gauge fields’. In simple terms, a gauge field is a recipe saying how a particle transforms as we move it along a path in spacetime. Formalizing this brings in groupoids and their connection to group theory, topology and geometry.

To see how, note that ‘spacetime’ is really just an example of a space, where we treat time as an additional dimension. Thus we can associate to spacetime a groupoid in the manner discussed, where the objects are points and the morphisms are equivalence classes of paths. Particles, on the other hand, have symmetries of a geometrical nature, and these symmetries form a group. A gauge field is a map from the groupoid associated to spacetime to the group of symmetries of a particle.

For another example, we can reconsider the equation 2 + 3 = 5. This takes a whole new appearance in the light of category theory. We can see it as summarizing an isomorphism in the category of finite sets. Understanding this requires that we think about ‘decategorification’, which is the process of treating isomorphisms as equations.

All this is quite worthy of thought, but mathematicians have by now gone much further in their investigation of sameness. Category theory is just the tip of an iceberg, namely the theory of infinity-categories. The key idea is to stick firmly to the intuition that sameness is more precisely described by specifying an isomorphism than by merely asserting equality. Thus, instead of talking about whether two morphisms are equal or not, we should go ahead and talk about isomorphisms between them. For this we can use a structure with objects, morphisms between objects, 2-morphisms between morphisms, and so on ad infinitum. This is called an ‘infinity-category’.

Examples are not hard to come by. In topology they arise starting from a space by considering points, paths, ‘paths of paths’ (that is, one-parameter families of paths), and so on. This particular kind of infinity-category is an ‘infinity-groupoid’, meaning that the morphisms at all levels are invertible up to morphisms at the next higher level.

Infinity-groupoids also play an important role in string theory: they give higher generalizations of groups, which serve to describe symmetries in ‘higher gauge theory’. The reason is that just as ordinary groups are used to describe how particles transform when moved along paths, higher groups describe how strings are moved along paths of paths, and so on.

In the last decade or so, there has been an interesting shift in attitudes regarding the role of equality in infinity-category theory. For quite a while, it seemed natural in certain circles that we should ban talk of equality whenever possible, speaking instead of isomorphisms, or more precisely, ‘equivalences’. Michael Makkai’s ‘first-order logic with dependent sorts’ makes it possible to enforce such a ban within the foundations of mathematics. The consequences are still just beginning to be explored.

However, around 2005, Vladimir Voevodsky, building on previous work, began promoting the ‘axiom of univalence’. This says, briefly, that ‘equivalence is equivalent to equality’. This axiom has the effect of rehabilitating equality. Indeed, one might say it effectively redefines equality to mean equivalence! There is now a powerful program, called ‘homotopy type theory’, seeking to redo the foundations of mathematics based on this idea.

There are several ways to think about the axiom of univalence. One can see it as a sophisticated updating of Leibniz’s principle of the identity of indiscernibles. However, a full understanding of this axiom requires understanding how the foundations of mathematics become infected by topology when we try to base mathematics on infinity-groupoids. Sets are re-envisioned as a special case of spaces, and the axiom of univalence states a property of the ‘space of all spaces’.

One goal of this paper is to explain this in simple terms. Another, more general goal is to show how the study of equality, isomorphism, equivalence and other notions of sameness leads to fruitful new interactions between logic, topology, and physics, which make rich topics for philosophical inquiry.

## Re: Categories for the Working Philosopher

You work at another UC and you still can’t tell Irvine from Davis? They’re 400 miles apart.