presets and monoids

presets and monoids

[Vasili I. Galchin]

Hello Cat Group,

       I have a question about adjoints.

       Lets consider presets P and P'. If we have adjoints between P
and P' .. say (A, A'), what are the unit and co-unit natural
transformations, respectfully?

       Likewise lets consider monoids M and M'. If we we have adjoints
between M and M',  (A, A') ... .what are the unit and co-unit natural
transformation equations, respecfully?

Regards,

Vasili


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Re: presets and monoids

[Fred E.J. Linton]

On Wed, 24 Oct 2012 07:37:49 PM EDT "Vasili I. Galchin" <vigalchin@gmail.com>
asked:

>        Lets consider presets P and P'. If we have adjoints between P
> and P' .. say (A, A'), what are the unit and co-unit natural
> transformations, respectfully?

If "presets" means pre-ordered sets (what I'd prefer to call just posets)
then those natural transformations are just the order relations
p </= a'a(p) (unit) and aa'(p') </= p' (co-unit) .

>        Likewise lets consider monoids M and M'. If we we have adjoints
> between M and M',  (A, A') ... .what are the unit and co-unit natural
> transformation equations, respecfully?

If "respecfully" means "respectively", and if "monoid" means "category
with one object", then the behavior of the functors A and A' is forced
at the object level (each carries the only available object to the only 
available object), and the adjointness (M(*, A'*') = M'(A*, *')) forces  
M and M' to be isomorphic as monoids.

On the other hand, if my guesses as to the intended meanings of those 
undefined terms are in error, then all bets are off :-) .

> Regards,
> 
> Vasili

HTH. Cheers, -- Fred



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