I suppose you've noticed that, with C the category of T_2 spaces, and I a 1-point space, your "point cardinality" of the real line (usual topology) becomes "alef-nought"? Is that OK by you? Cheers, -- Fred -- ------ Original Message ------ Received: Wed, 23 Mar 2011 06:39:33 PM EDT From: Aleks Kissinger <aleks0@gmail.com> To: categories <categories@mta.ca> Subject: categories: Is this a studied notion of cardinality?
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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