Subject: categories: Re: cracks and pots

From: Steve Vickers <[email-masked]>
Date: Sun, 19 Mar 2006 18:25:08 +0000

On 17 Mar 2006, at 09:36, George Janelidze wrote:
> ... I think if we really care about
> relations between category theory and "other foundational
> disciplines", we
> should begin by explaining that category theory is not just a language
> allowing one to call homology a functor, but that category theory has
> beautiful constructions and results (some already from 1940s and 50s!)
> making enormous simplifications/applications/illuminations in
> neighbour
> areas of pure mathematics, such as abstract algebra, geometry, and
> logic.

Dear George,

I think the straight answer is that it is genuinely difficult.

Even for elementary applications it is not easy. Try asking non-
categorical topologists how they explain the product topology to
students. Many will say, "This definition may look odd, but it turns
out to work best." Others will produce various ad hoc justifications,
such as "It's the definition that makes Tychonoff's theorem
true." (Though that may be at least historically correct.)   You
point out that the product topology is the unique one such that
projections are continuous and tupling preserves continuity,  but
they still don't see that as anything special.

But with regard to certain advanced applications, there are pictures
in the minds of the category theorists that do not translate at all
easily to paper. Even the master expositors find it hard. I'm
thinking for example of the idea of topos as generalized space.

I have been working seriously with toposes (usually as generalized
spaces) for about 15 years now and in some respects my understanding
of them is quite deep. Yet there is still a huge gap in my
understanding when it comes to their applications in algebraic
geometry, Galois theory and algebraic topology, the kind of fields
that gave rise to toposes in the first place. Somehow when I read the
accounts I see a mass of machinery but no clear intuitions for what
it is doing. This surprises me. A characteristic strength of category
theory is that it is particularly good at explaining the underlying
meaning of constructions, with its notion of universal properties,
and with some beautiful tricks of categorical logic.

So is it possible to explain, or illuminate, those particular
categorical applications to someone like me? (Perhaps the challenge
has already been met, and I've just missed the right book; and of
course I eagerly await vol. 3 of the Elephant.)

Here's a sample question where my categorical understanding falls
short of the applications.

If A is an Abelian group, then the space ^A of A-torsors is also a
group (modulo canonical isomorphisms - the equational laws of group
theory do not necessarily hold up to equality). The identity element
is the regular representation of A on itself, group multiplication is
"tensor product" of torsors, and inverses are got by inverting the A-
action.

It follows that if X is any space, then the collection of continuous
maps from X to ^A is also a group, and this construction is self-
evidently contravariant in X.

Obviously it takes topos theory to formalize this, but already we can
paint a picture.

For example, suppose A is the cyclic group C_2 of order 2 and X is
the circle. Then there are (classically) two isomorphism classes of
maps T: X -> ^A, essentially because in going once right round the
circle the variable torsor T(x) can come back either just how it
started or with an automorphism swapping its two elements. The
corresponding group is C_2.

This looks like some kind of cohomology, so is it already part of the
standard theory? I've never managed to follow all the machinery through.

All the best,

Steve Vickers