Re: Is this a studied notion of cardinality?

[soloviev@irit.fr]

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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