Unpacking the Objects in the Category of Sets: Cardinality and Distinctions

[AgentBased]

Can someone give a quick description of the objects in the category SET? In particular, are sets distinguished by anything more than cardinality (i.e. R^2 has the same cardinality as R--are they distinct objects in SET, or is there just one "uncountable set" object?)

Answers/help much appreciated!

 

[Hurkyl]

There is a one-to-one correspondence between (small) sets and objects in Set. So R and R² are, indeed, distinct but isomorphic objects.

Set, of course, is equivalent to its skeleton, whose objects are (small) cardinal numbers. And c=c². (I'm using c for the cardinality of R)

Many constructions with categories are only defined up to equivalence -- so you can often replace Set with its skeleton when its convenient to do so.



P.S. "uncountable" means any cardinality greater than $\aleph_0$. I'm assuming you really meant just the cardinality of R

 

[AgentBased]

Thanks! (You are right, of course, I was being sloppy in using the term "uncountable" as I did.)

So suppose you had a function in SET, something like R^2 -> R^2 given by (x_1,x_2) -> (x_1,0). Would this arrow actually go from R^2 to itself, or to some distinct set object indicating the subset? (I am trying to determine idempotent arrows in SET, without being too blatant about it.)

 

[Hurkyl]

R^2 -> R^2 ... Would this arrow actually go from R^2 to itself

Yes -- that's what "R² -> R²" means.


There are a variety of algebraic manipulations you can do on a function -- things like invoke the existence of an epic-monic factorization, or use it to define an adjoint pair (inverse image, direct image) of functors on the poset Sub(R²) of subobjects of R². (Set is nice enough to allow these constructions -- but they don't work in bad categories) I'm not really sure what you're looking for.



It almost sounds like you are talking about a construction I saw in Categories, Allegories by Freyd and Scedrov: they define a category Split(E) whose objects are the idempotents of E and Hom(e, e') consists of all morphisms of E satisfying xe = x = e'x.

Oh! Wikipedia has an article on it here.

For each idempotent in the category, this construction formally adds a new object representing its image. (which is named by the idempotent itself) Set already has images, though, so Split(Set) turns out to be equivalent to Set.

 

[AgentBased]

Thank you! I have some background in linear algebra and group theory, but I am just starting categories. I am trying to do some independent study in Categories for the Working Mathematician, and sometimes just a little clarification or explication helps so much when working on the exercises.