Re: Some books on category and topos theory

From: Tim May <tcmay.domain.name.hidden>
Date: Tue, 9 Jul 2002 09:24:54 -0700

On Tuesday, July 9, 2002, at 07:41 AM, Bruno Marchal wrote:
>

> Tim makes a very genuine remark (but he writes so much I fear that has
> been unnoticed!).

True enough...I write a lot! (The old joke applies: "I don't have enough
time to write a short letter.")


> He said: read Tegmark (Everything paper), then learn
> category, then read again Tegmark.

Well, I didn't actually say "then learn category (theory). I said spend
enough time looking at category theory to get the gist of what they are.
A couple of days, for example.

Reading styles differ, but I have come to favor the "hawk spiral." I see
hawks spiralling in the thermals near my house, and this is how I like
to learn. I read something from one book, think, read from another,
think, try to compare what the authors are saying, read from another, go
back to the earlier book and read more, and so on. A helix, covering the
same material many times.

> Indeed I would say category theory has
> emerged from the realisation that mathematical structures are themselves
> mathematically structured. Categorist applies the every-structure
> principle
> for each structure. Take all groups, and all morphism between groups:
> you
> get the category of groups. It is one mathematical structure, a category
> (with objects = groups and arrows = homomorphism) which, in some sense
> capture the essence of group.

Exactly. A very nice explanation.

And much of what Tegmark outlines in his large chart can be dramatically
simplified and abstracted. Crane, Baez, Dolan, and others call this the
"categorification" process. Robert Geroch's textbook, "Mathematical
Physics," uses categories and functors throughout as a unifying (and
intuition-increasing) tool.

Hey, let me be very clear about something: I don't know what the
categorification of Tegmark's ideas are!

Categories and toposes are not a magic bullet.

But I know that gettting lost in the swamps of mathematical structures
is a real danger, and that mathematicians have found certain unifying
symmetries, structures, parallels which simplify things dramatically.
Category theory is a lot like finding metaphors and parallels. (We use
the term "isomorphism" almost in everyday language, so the leap to all
sorts of morphisms is not great.)

>
> Categories arises naturally when mathematician realised that many proofs
> looks alike so that it is easier to abstract a new
> structure-of-structures,
> then makes proofs in it, then apply the abstract proof in each structure
> you want. So they define universal constructions in category (like the
> "product"), which will correspond automatically to
> - "and" in boolean algebra
> - "and" in Heyting algebra
> - group product in cat of groups
> - topological product in cat of topological spaces
> - Lie product in cat of Lie groups, etc.
> So Category theory helps you to make a big economy of work ... once you
> invest in it, if you are using algebra. It saves your time.

Exactly. Another good explanation here.

And it's more than just a notational convenience. Proofs in one area,
such as some branch of topology, can be transformed into proofs in other
areas.

(Aside: I believe this is a big part of what thinking is about: applying
thoughts/concepts/morphisms/etc. from one area to another. Perhaps
category theory will push AI in new ways. Perhaps the "frame problem"
will be solvable with new tools.)

> But, to come back to Tim remark, it hints that a giant part whole of
> mathematics is naturally mathematically structured, and this should be
> taken
> into account.

I first heard of category theory about 10 years ago. A friend of mine
was working for a company in Palo Alto which was using category theory
to model economic data bases (such as petroleum reserves, ports in
different countries, etc. ...very probably CIA-related, now that I think
about it). He didn't have the interest or insight to explain why
category theory was so cool.

I asked a mathematician friend of mine (Eric H., for Hal and WD) about
it and he said it was about what mathematicians do when they draw
diagrams on blackboards. It didn't sound very interesting. It sounded
like some variant of denotational semantics.

But when the light bulbs went off this spring, when I dug into the
writings of Baez, Hillman, Markopoulou, and the books of Lawvere, Mac
Lane (difficult), and others, I had a major epiphany, a real "Ah ha!"
experience.

Everything seemed directly related to problems which had fascinated me
for decades. Some of these issues I hope I have hinted at here.

It was almost as if category and topos theory had been invented just for
me....an exaggeration, but it captures my sense of wonder. I haven't
been this excited about a new area in more than a decade. I expect I'll
be doing something in this area for at least the _next_ decade.

My apologies if this explanation of enthusiasm is too personal for you
the reader, but I think enthusiasm is a good thing.


--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Received on Tue Jul 09 2002 - 09:26:09 PDT