Topos Theory in a Nutshell

John Baez

October 10, 2021

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. Finally I’ll tell you some more
things to read.

I’ll warn you: it takes a lot of work to learn enough topos theory to
really use it to solve problems. Thus, when you’re getting
started the main reason to learn about it should not be to quickly
solve some specific problems, but to broaden your horizons and break
out of the box that traditional mathematics, based on set theory,
imposes on your thinking.

1. Hand-Wavy Vague Explanation

Around 1963, Bill 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.

In the spring of 1966 Lawvere encountered the work of Alexander
Grothendieck, who had invented a concept of “topos” in his work on
algebraic geometry. The word “topos” means “place” in Greek. In
algebraic geometry we are often interested not just in whether or not
something is true, but in where it is true. For example, given
two functions on a space, where are they equal? Grothendieck thought
about this very hard and invented his concept of topos, which is
roughly a category that serves as a place in which one can do
mathematics.

Ultimately, this led to a concept of truth
that has a very general notion of “space” built into it!

By 1971, Lawvere and Myles Tierney had taken Grothendieck’s original
concept of topos — now called a “Grothendieck topos”
— generalized and distilled it, and come up with the concept
of topos I’ll talk about here. This is sometimes called an
“elementary topos”, to distinguish it from Grothendieck’s
notion… but often it’s just called a topos, and that’s what
I’ll do.

So what is a topos?

A topos is category with certain extra properties that make it a lot
like the category of sets. There are many different topoi; you can do
a lot of the same mathematics in all of them; but there are also many
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. The reason is that truth is not a yes-or-no
affair: instead, we keep track of “how” true statements are, or more
precisely where they are true. Some but not all topoi contain
a “natural numbers object”, which plays the role of the natural
numbers.

But enough hand-waving. Let’s see precisely what a topos is.

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.

It’s not too long! But it could be made even shorter: we don’t need to
mention colimits, since that follows from the rest.

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:

• an initial object (an object like the empty set)
• a terminal object (an object like a set with one element)
• binary coproducts (something like the disjoint union of two sets)
• binary products (something like the Cartesian produc