Topos Theory in a Nutshell

John Baez

April 12, 2006

Okay, you wanna know what a topos is? First I'll give you a hand-wavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples.

I'll warn you: despite Chris Isham's work applying topos theory to the interpretation of quantum mechanics, and Anders Kock and Bill Lawvere's work applying it to differential geometry and mechanics, topos theory hasn't really caught on among physicists yet. Thus, the main reason to learn about it is not to quickly solve some specific physics problems, but to broaden our horizons and break out of the box that traditional mathematics, based on set theory, imposes on our thinking.

1. Hand-Wavy Vague Explanation

Around 1963, Lawvere decided to figure out new foundations for mathematics, based on category theory. His idea was to figure out what was so great about sets, strictly from the category-theoretic point of view. This is an interesting project, since category theory is all about objects and morphisms. For the category of sets, this means SETS and FUNCTIONS. Of course, the usual axioms for set theory are all about SETS and MEMBERSHIP. Thus analyzing set theory from the category-theoretic viewpoint forces a radical change of viewpoint, which downplays membership and emphasizes functions.

Even earlier, this same change of viewpoint was also becoming important in algebraic geometry, thanks to the work of Grothendieck on the Weil conjectures. So topos theory can be thought of as a merger of ideas from geometry and logic - hence the title of this book, which is an excellent introduction to topos theory, though not the easiest one:

After a bunch of work, Lawvere and others invented the concept of a "topos", which is category with certain extra properties that make it a lot like the category of sets. There are lots of different topoi; you can do a lot of the same mathematics in all of them; but there are also lots of differences between them: for example, the axiom of choice need not hold in a topos, and the law of the excluded middle ("either P or not(P)") need not hold. Some but not all topoi contain a "natural numbers object", which plays the role of the natural numbers.

It's good to prove theorems about topoi in general, so that you don't need to keep proving the same kind of theorem over and over again, once for each topos you encounter. This is especially true if you do a lot of heavy-duty mathematics as part of your daily work.

2. Definition

There are various equivalent definitions of a topos, some more terse than others. Here is a rather inefficient one:

A topos is a category with:

A) finite limits and colimits,

B) exponentials,

C) a subobject classifier.

Short and sweet! But it could be made even shorter.

3. Some Consequences of the Definition

Unfortunately, if you don't know some category theory, the above definition will be mysterious and will require a further sequence of definitions to bring it back to the basic concepts of category theory - object, morphism, composition, identity. Instead of doing all that, let me say a bit about what these items A)-C) amount to in the category of sets:

A) says that there are:

In fact A) is equivalent to all this stuff. However, I should emphasize that A) says all this in an elegant unified way; it's a theorem that this elegant way is the same as all the crud I just listed.

B) says that for any objects x and y, there is an object yx, called an "exponential", which acts like "the set of functions from x to y".

C) says that there is an object called the "subobject classifier" Ω, which acts like {0,1}, in that functions from any set x into {0,1} are secretly the same as subsets of x. You can think of Ω as the replacement for the usual boolean "truth values" that we work with when doing logic in the category of sets.

Learning more about all these concepts is probably the best use of your time if you wants to learn a little bit of topos theory. Even if you can't remember what a topos is, these concepts can help you become a stronger mathematician or mathematical physicist!

4. Examples

Suppose you're an old fuddy-duddy. Then you want to work in the topos Set, where the objects are sets and the morphisms are functions.

Suppose you know the symmetry group of the universe, G. And suppose you only want to work with sets on which this symmetry group acts, and functions which are compatible with this group action. Then you want to work in the topos G-Set.

Suppose you have a topological space that you really like. Then you might want to work in the topos of presheaves on X, or the topos of sheaves on X. Sheaves are important in twistor theory and other applications of algebraic geometry and topology to physics.

Generalizing the last two examples, you might prefer to work in the topos of presheaves on an arbitrary category C, also known as hom(Cop, Set).

For example, if C = Δ (the category of finite totally ordered sets), a presheaf on Δ is a simplicial set. Algebraic topologists love to work with these, and physicists need more and more algebraic topology these days, so as we grow up, eventually it pays to learn how to do algebraic topology using the category of simplicial sets, hom(Δop, Set).

Or, you might like to work in the topos of sheaves on a topological space - or even on a "site", which is a category equipped with something like a topology. These ideas were invented by Alexander Grothendieck as part of his strategy for proving the Weil conjectures. In fact, this is how topos theory got started. And the power of these ideas continues to grow. For example, in 2002, Vladimir Voevodsky won the Fields medal for cracking a famous problem called Milnor's Conjecture with the help of "simplicial sheaves". These are like simplicial sets, but with sets replaced by sheaves on a site. Again, they form a topos. Zounds!

But if all this sounds too terrifying, never mind - there are also examples with a more "foundational" flavor:

Suppose you're a finitist and you only want to work with finite sets and functions between them. Then you want to work in the topos FinSet.

Suppose you're a constructivist and you only want to work with "effectively constructible" sets and "effectively computable" functions. Then you want to work in the "effective topos" developed by Martin Hyland.

Suppose you like doing calculus with infinitesimals, the way physicists do all the time - but you want to do it rigorously. Then you want to work in the "smooth topos" developed by Lawvere and Anders Kock.

Or suppose you're very concerned with the time of day, and you want to work with time-dependent sets and time-dependent functions between them. Then there's a topos for you - I don't know a spiffy name for it, but it exists: an object gives you a set S(t) for each time t, and a morphism gives you a function f(t): S(t) → T(t) for each time t. This too gives a topos!

If you want to learn more about topos theory, this is the easiest place to start:

It may seem almost childish at first, but it gradually creeps up on you. Schanuel has told me that you must do the exercises - if you don't, at some point the book will suddenly switch from being too easy to being way too hard! If you stick with it, by the end you will have all the basic concepts from topos theory under your belt, almost subconsciously.

After that, try this one:

This is a great introduction to category theory via the topos of sets: it describes ordinary set theory in topos-theoretic terms, making it clear which axioms will be dropped when we go to more general topoi, and why. It goes a lot further than the previous book, and you need some more sophistication to follow it, but it's still written for the beginner.

I got a lot out of the following book, but many toposophers complain that it's not substantial enough - it shows how topoi illuminate concepts from logic, but it doesn't show you can do lots of cool stuff with topoi. Perhaps it's been supplanted by Sets for Mathematics, but you should definitely take a look at it if you can find it:

Don't be scared by the title: it starts at the beginning and explains categories before going on to topoi and their relation to logic.

When you want to dig deeper, try this:

It's still an introductory text, but of a more muscular sort than those listed above. McLarty is a philosopher by profession, but this is very much a math book.

To dig deeper still, try Mac Lane and Moerdijk's book mentioned above. And after that... well, let's not rush this! For example, this classic is now available for free online:

but it's advanced enough to make any beginner run away screaming! These books are bound to have a similar effect:

... but once you get deeper into topos theory, you'll see they contain a massive hoard of wisdom. I'm trying to read them now. McLarty has said that you can tell you really understand topoi if you can follow Johnstone's classic Topos Theory. It's long been the key text on the subject, but as a referee of his new trilogy wrote, it was "far too hard to read, and not for the faint-hearted". His Sketches of an Elephant spend more time explaining things, but they're so packed with detailed information that nobody unfamiliar with topos theory would have a chance of seeing the forest for the trees. Also, they assume a fair amount of category theory. But they're great!

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. - Hermann Weyl

© 2006 John Baez