re: Small is beautiful

[John Power <A.J.Power@bath.ac.uk>]

Dear Colleagues,

I have not quite absorbed all the email on this yet, so may be  
repeating something already said. But perhaps it would be helpful to  
mention that, in regard to questions like this, I have found enriched  
categories helpful:

consider either

1 the functor category [->,Set] (an object is a pair of sets X and Y  
and a function from X to Y)

or

2 the category Sub(Set) (an object is a set X together with a subset  
X', and a map from (X,X') to (Y,Y') is a function from X to Y for  
which the image of X' lies in Y'

These categories, especially the first, both have the properties one  
typically seeks for a V in studying V-categories.

Spelling out what a V-category is in the second case yields a category  
C with a subcategory for which the inclusion is the identity on objects.

Happy New Year to all,

John.


Quoting Robert Pare <pare@mathstat.dal.ca>:

>
> I would like to add a few thoughts to the "evil" discussion.
>
> My 30+ years involvement with indexed categories have led me
> to the following understanding. There are two kinds of categories,
> small and large (surprise!). But the difference is not mainly one
> of size. Rather it's how well we can pin down the objects. The
> distinction between sets and classes is often thought of in terms
> of size but Russell's problem with the set of all sets was not one of
> size but rather of the nature of sets. Once you think you have the set
> of all sets, you can construct another set which you had missed.
> I.e. the notion is changing, slippery. There are set theories where
> you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)
> Smallness is more a question of representability: a functor may fail to
> be representable because it's too big (no solution set) or, more often,
> because it's badly behaved (doesn't preserve products, say). Subfunctors
> of representables are not usually representable.

...


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]