Is this a studied notion of cardinality?

[Aleks Kissinger <aleks0@gmail.com>]

Let C be a category with a chosen "point" object I (i.e. tensor unit).
The "point cardinality" of some object X in C is then the minimum
number of points "p : I --> X" required to distinguish any two maps
f,g : X --> Y for any Y. Supposing all objects even have a point
cardinality implies well-pointedness of the category, but can actually
be quite a bit stronger, if in general the point cardinality is much
less than | hom(I,X) |.

Of course, the thing I have in mind here is dimension of a vector
space, where N points are picking out N basis vectors. So, my
questions are:
1. is point-cardinality the the most natural generalisation of this notion?
2. does it provide useful information in categories that are bit like
vector spaces, like projective spaces or certain kinds of modules of
an algebra?


Aleks


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