Re: universal element for the functor

In article
<15793250.1141316667274.JavaMail.jakarta@xxxxxxxxxxxxx
forum.org>,
Meg Weiss <megweiss@xxxxxxxxx> wrote:

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A universal element for a functor F is an ordered
pair (u,R)
consisting of a set R and an element u \in F(R) with
following
property:

I guess your functor goes from Set to some category
whose objects are
sets and whose arrows are functions on the underlying
sets...

To any set S and any element s \in F(S) there is
exactly one function
h: R -> S with F(f)(u) = s.

Let the functor F be F(S) = S x S, F(f) = f x f.

1) For 2 = {1, 2}, prove that (1,2) \in 2 x 2 is a
universal element
for the functor F.

Are you asking us to do your homework for you? To
explain the
question? To give a hint? What?

This is not a homework.




What have you managed? What are you confused about?


I wrote down
F(2) = {(1,1), (1,2), (2,1), (2,2)}
and try to define f as
f(1)= 1, f(2)=2
then I have problem.

Since we need a functor to be satisfied such that
F(f)(1) = (1,2) etc.
But
F(f)(1) = (f x f)(1) = f(1)xf(1) = (1,1)
F(f)(2) = (f x f)(2) = f(2)xf(2) = (2,2)

I don't get (1,2) where it should be.
How should I define f?


2) For 3 = {1, 2, 3}, prove that (1,2) \in 3 x 3 is
not a universal
element for the functor F


3) For 1 = {1}, prove that (1,1) \in 1 x 1 is not
universal for F.

Likewise.


--
======================================================
================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

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