coordinate systems, variables w/ categories?

analytic_at_gmail.com
Date: 02/11/05 

Date: 11 Feb 2005 13:48:32 -0800

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.

(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 defining a category C whose objects are mappings (functions from R->R
say), but whose objects are formal variables x_1,x_2,... representing
different coordinate systems I guess. (or rather, the objects in the
category are generated by the "x_i"s and their products)

There is another category D with the same objects (x_1,...,x_n) (though
I don't think products are necessary this time), but with the morphisms
representing coordinate changes, and your route to changing coordinate
systems shouldn't matter - i.e. given two routes (morphisms) from x_i
to x_j, they should be equal, or more categorically, say that
hom(x_i,x_j) has at most one element. Denote this function <i,j>, if
it exists.

As always, the diagonal function dn:x_i->(xi,xi,...,xi) (that's n
copies)

Now, given f(x_a1,x_a2,...,x_an)->x_l, define it's evaluation at a
variable x_i, if it exists to be

f|i:R->R
   :x|-->(<a1,i>,<a2,i>,...,<an,i>)dn(x)

Now, for each variable x_i, use something like a forgetful partial
functor
    U(x_i):C->hom(R,R)
        :x_j |--> <j,i>
        :f |--> f|i (when defined).

i.e., say that two functions f and g in C are equal as functions if f|i
and g|i are both defined for some i, and then f|i=g|i.

It seems to me that in any (at least concrete) category, taking any
composite function (or more generally any set of composites), say
fgh,that you can embed it into C by assigning f g and h to different
hom-sets in C, such that the composite fgh is still defined (I am aware
that if it was something like fgfh, not that the first f would not be
mapped to the the same element as the second f). Also, in this case,
the natrual U(i) to use would be U(the codomain of the leftmost
function), i.e. U(cod(f)) in the case of fgh..

In this case (where . is composition, and x is multiplication (suitably
defined)), viewing the differential operator as preserving the domain
and range of morphisms in C, and embedding f.g into C (By some suitably
complex bunch of functors), you get that D(f.g)=D(f)xD(g) (as functions
forgotten from U), right? So you can regard D as some weird functor
from hom(R,R) to mul(R,R) (where [fg](x)=f(x)xg(x))

So, any points to any pleasant topics that might be related to this?

thanks, (sorry if this is a bit messy, but I thought to ask before.
Also, I haven't had a chance to check the functoriality (or lack
thereof) of anything yet))

Stephen