Re: Is this a studied notion of cardinality?

Re: Is this a studied notion of cardinality?

[Fred E.J. Linton]

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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Re: Is this a studied notion of cardinality?

[soloviev]

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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Re: Is this a studied notion of cardinality?

[Aleks Kissinger]

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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Is this a studied notion of cardinality?

[Aleks Kissinger]

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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