Dear Alex, I had some papers where the fact that tensor unit I is a generator (i.e., your "point cardinality" is 1) is used to describe ALL natural transformations between superpositions of distinguished functors (for example, tensor and internal hom in symmetric monoidal closed categories, or in compact closed categories) S.V. Soloviev. On natural transformatioms of distinguished functors and their superpositions in certain closed categories.-J.of Pure and Applied Algebra 47(1987) p.181-204. or of superpositions of tensor and biproduct Robin Cockett, Martin Hyland, Sergei Soloviev. Natural transformation between tensor powers in the presence of direct sums. Rapport de recherche, 01-12-R, IRIT, Universit´e Paul Sabatier, Toulouse, juillet 2001. The technique can be used in case of "multiple generators" (your "point cardinality" > 1) but was never detailed as a paper. This is about possible applications of the notion you suggest. Regards Sergei Soloviev
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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