Poset adjoints

Poset adjoints

[Vasili I. Galchin]

Hello Category Community,

       Given that poset adjoints are considered miniature versions of
adjoints, what are are the unit and co-unit natural transformations
say between poset A and B?

Kind regards,

Vasili


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Re: Poset adjoints

[Harley D. Eades III]

Hi, Vasili.

On Apr 16, 2014, at 1:51 AM, Vasili I. Galchin <vigalchin@gmail.com> wrote:

> Hello Category Community,
>
>       Given that poset adjoints are considered miniature versions of
> adjoints, what are are the unit and co-unit natural transformations
> say between poset A and B?

Doesn't this boil down to a Galois connection?  The unit and counit would
then be:

f(y) <= x iff y <= g(x)

where f : B -> A and G : A -> B and x in A and y in B.

Very best,
.\ Harley

>
> Kind regards,
>
> Vasili
>


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Re: Poset adjoints

[Fred E.J. Linton]

Properly posed, Vasili's question should be not

>        Given that poset adjoints are considered miniature versions of
> adjoints, what are are the unit and co-unit natural transformations
> say between poset A and B?

but:
  
: Given order-preserving functions r: A --> B and l: B --> A between
: posets A and B, with r right adjoint to l, what are the unit and 
: co-unit natural transformations 1_B ==> rl and lr ==> 1_A ?

And the answer, as always, is that 1_B: b --> rlb (for b in B) is 
the order relation b < rlb that adjointness correlates to lb < lb, 
while 1_A: lra --> a (for a in A) is the order relation lra < a 
that adjointness correlates to ra < ra.

(Above I'm writing simply < for the (reflexive, transitive) order relation
on either of the two posets.)

HTH. Cheers, -- Fred




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