I was very impressed by the description of SymPy here:

http://www.euroscipy.org/presentations/slides/index.html

and especially slide 14 "Internals: Object oriented model".

I have a couple of general questions about how Object-oriented SymPy
is. I'm a mathematician and would like to use or extend SymPy to do
abstract differential geometry.

1. Where/how can I see a diagram of the structure of the classes in
SymPy?
2. Is SymPy at all modeled after category theory, which organizes
mathematics in an essentially object-oriented way?
3. Is there a SymPy approach to, for example, vector spaces?  I do
*not* mean matrix algebra or computations in components.  I mean an
actual abstract vector space, where objects would be vectors and one
could take linear combinations of vectors.  I'm asking this question
as an example of a more general question - whether SymPy has been used
to do "abstract" computations on mathematical objects such as vector
spaces, manifolds, etc. (as mathematicians often do) rather than
explicit ones in numbers or polynomials.

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