Re: Is this a studied notion of cardinality?

["Fred E.J. Linton" <fejlinton@usa.net>]

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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