# Objects in the category SET?

1. Dec 25, 2009

### 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?)

2. Dec 25, 2009

### Hurkyl

Staff Emeritus
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

3. Dec 26, 2009

### 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.)

4. Dec 26, 2009

### Hurkyl

Staff Emeritus
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.

5. Dec 27, 2009

### 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.