Subject: categories: Modeling infinitesimals with 2x2 matrices

From: "John Baez" <[email-masked]>
Date: Wed, 28 Apr 2004 17:54:01 -0700 (PDT)

Vaughan Pratt writes:

> Why not model d as the matrix
>
> 0 1
> 0 0 ?
>
> This is a perfectly good quantity, adding and scaling just like any
> real, e.g.
>
> 2d =  0 2
>       0 0.
>
> And obviously d^2 = 0.

Part of this idea is implicit in the usual algebraic geometry treatment
of infinitesimals as nilpotents.  In addition to the usual "point", such
that complex functions on this space form the commutative ring C, algebraic
geometers like to think about the "point with nth-order nilpotent fuzz",
such that complex functions on this space form the commutative ring
C[d]/<d^n = 0>.   They visualize this as a space slightly bigger than
a point: just big enough to tell the difference between the function 0
and the function whose first n-1 derivatives equal zero!

To deal with this sort of "space" in a precise way, someone like Grothendieck
invented the category of affine schemes, which is just the opposite of the
category of commutative rings.  But affine schemes are happier as part of a
larger category of schemes... and thus topos theory was brought kicking
and screaming into the world.  To see how this led to a really nice treatment
of infinitesimals, see:

F. William Lawvere, Outline of synthetic differential geometry, available
at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html

or

Anders Kock, Synthetic Differential Geometry, Cambridge U. Press,
Cambridge, 1981.

But, it's also tempting to embed the commutative ring C[d]/<d^n = 0> into
the noncommutative ring of nxn complex matrices, by letting d be a
slightly off-diagonal matrix, like this:

0 1 0 0
0 0 1 0                    (in the case n = 4)
0 0 0 1
0 0 0 0

(Vaughan is considering the case n = 2.)  And this is more like how
Alain Connes thinks of infinitesimals: as part of the bigger world of
noncommutative geometry!

Best,
jb