Re: coordinate systems, variables w/ categories?

From: Larry Hammick (larryhammick_at_OMIT-MEtelus.net)
Date: 02/12/05 

Date: Sat, 12 Feb 2005 02:14:34 GMT

<analytic@gmail.com>
> Hi,
>
> A friend of mine was terribly confused today - his quoted one of his
> lecturers as saying that the derivative of a composite is equal to the
> composite of the derivatives or some such thing.

That is true, if it is understood that the derivative _at a point_ is a
linear mapping from the one Banach space to the other. (That's how the
derivative of a function of more than one variable is nowadays defined. The
old way used a matrix of partial derivatives, with formulas to describe how
the coefficients change with a change of coordinates.)

> (I'm guessing the following comes up in geometry, but I haven't come
> across it before - actually, what I'm really asking is for someone to
> point me in a direction where I can read more on the topic).
>
> I've thought about those notions before, and managed to work out a
> rough context in which that made sense, namely in a category that has a
> rough notion of "coordinate systems".

I'm not fluent in category theory, but it sounds to me like you're after
"manifolds".
LH

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