From: baez@galaxy.ucr.edu (John Baez)
Subject: Topos theory for physicists
Date: 28 Dec 2000 19:12:47 GMT
Newsgroups: sci.physics.research
Summary: [missing]
In article <925bvt$6as$1@nnrp1.deja.com>, wrote:
Chris Hillman asked:
>> Have you read the book by Moerdijk and Mac Lane, Sheaves in Geometry
>> and Logic: A First Introduction to Topos Theory, Springer, 1992?
>Nope, have to try it out. Actually, I heard the word topos a rather
>large number of times and still don't know what it is! This begins to
>be annoying, if you know what I mean... :-)
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 relevant to physics.
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 MacLane and Moerdijk's book.
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 that has:
A) finite limits and colimits
B) exponentials
C) a subobject classifier
Now, if you don't know a little category theory, this 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 of morphisms, identity morphism. 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 an object like a set with one element)
binary coproducts (something like the disjoint union of two sets)
binary products (something like the Cartesian product of two sets)
equalizers (something like the subset of X consisting of all elements
x such that f(x) = g(x), where f,g: X -> Y)
coequalizers (something like the quotient set of X where two elements
f(y) and g(y) are identified, where f,g: Y -> X)
[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 y^x, 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" Omega,
which acts like {0,1}, in that functions from any set x into {0,1} are
secretly the same as subsets of x.
Learning more about all these concepts is probably the best use of ones
time if one wants to learn a little bit of topos theory. For example,
recognizing a coequalizer when you see one can be very handy.
3. 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(C^{op},Set).
For example, if C = Delta (the category of finite totally ordered sets),
a presheaf on Delta 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(Delta^{op},Set).
There are also examples of 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".
Suppose 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.
In short, topos theory opens up a whole new world of beautiful
mathematics, which physicists will eventually fall in love with,
even though most haven't yet.