Fred: Yes! In (Vect, k) and (Haus, 1), this number identifies the cardinality of an object's "dense" subset, for the correct notion of "dense" in either category. More precisely, for ordinary (non-enriched) categories, the choice of a point object I gives a canonical choice of forgetful functor U := hom(I, --) into Set. Then, point cardinality is the set cardinality of the smallest set Q that satisfies the usual unique lifting condition: for e : Q --> UX, f : Q --> UY, if there exists a map g : X --> Y such that f = (Ug)e then g is unique. From this point of view, it seems Vect is a bit of a degenerate example, because all vector spaces are free over some set and unique lifting is just the universal property of the right adjoint U. David: Hmm... Yes, it seems like there's a lot of choices here on how you can use distinguishing points to pull out invariants of the category or objects in the category. Here's one: for a category C, form a category C+ whose objects are minimal sets of points { p : I --> X } and whose arrows S --> S' are maps f in C s.t. for p in S, f o s is in S'. This category C+ has fewer arrows but many more objects. There is also a canonical functor Q : C+ --> C that creates isos. The question is, does this have interesting properties? Take Z(C) and Z(C+) as the largest sub-groupoids of C and C+ respectively. Since Q creates isos, it restricts to an equivalence of categories Q' : Z(C+) --> Z(C). Z(C+) a new groupoid with "non-trivial" iso's unrolled. I.e. the only automorphisms in Z(C+) are permutations of distinguishing points. Aleks On 24 March 2011 00:29, Fred E.J. Linton <fejlinton@usa.net> wrote:

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 --

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