Re: topos and magic

[Vaughan Pratt <pratt@cs.stanford.edu>]

Andre Joyal wrote:
> I find the notion of elementary topos absolutly extraordinary, almost magical.
> Like every important mathematical discovery, it stands out as a colorful gem on a bed of grey stones.
> A classical example of gem is the field of complex numbers.

Historically these two gems emerged as entirely independent
developments.  However they are arguably facets of a single gem, the
abelian-topos categories Peter Freyd wrote about on this list in
November 1997, archived at
http://blog.gmane.org/gmane.science.mathematics.categories/day=19971031
or on Karel Stokkerman's topic-indexed archive at
http://www.mta.ca/~cat-dist/catlist/1999/atcat
and
http://www.mta.ca/~cat-dist/catlist/1999/prattsli

The complex numbers live within the ring M(2,R) of 2x2 real matrices as
a subring of M(2,R) that happens to form a field.  (The general linear
group GL(2,R) is a larger *skew* field in M(2,R), but is there a larger
*field* than the complex numbers therein?)  M(2,R) in turn forms a
one-object full subcategory of the abelian category Vct_R, with matrix
multiplication (hence complex number multiplication) realized as
composition.  So the complex numbers form a subcategory of an abelian
category.

As Peter's treatment makes clear, abelian categories are a quite minor
variant on toposes.  An abelian category (resp. topos) is an
abelian-topos category all of whose objects X are of type A (resp. T),
meaning that the first (resp. second) projection of Xx0, namely from Xx0
to X (resp. 0), is an iso.  This difference is expressed as a very
simple elementary (first-order) predicate whose intuitive meaning is
clear: simply multiply X by zero and see whether it remains X or
collapses to zero.

So within the relatively small universe of abelian-topos categories (by
comparison with *all* categories) we find lurking therein both the field
of complex numbers and the toposes (and hence in particular the
effective topos, yet another gem Andre mentioned).

Over on the Foundations of Mathematics mailing list, FOMers would
presumably connect complex numbers to set theory by observing that the
complex numbers form a set which lives within the universe of sets
axiomatized by the Zermelo-Fraenkel axioms.  To me the path from complex
numbers to toposes via matrices and abelian categories seems somehow
more intimate.  Simply calling the complex numbers a set seems dry as
dust (literally).

(Peter J., are abelian-topos categories in the Elephant?  They seem an
obvious candidate yet I couldn't find them in either the table of
contents or the index.)

Vaughan Pratt


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