Re: An elementary question

[alex <alexander.praehauser@gmx.at>]

I hope so much that I do not say something monumentally stupid. I am
just a student and do not want to attract negative attention, but is
there not, for any P, Q a whole partial order of "generalized
coproducts", in the sense that these consist of coproducts plus some
relations of the form (p \leq q) or vice versa, where these coproducts
are ordered by saying that one "generalized coproduct" is "smaller" than
the other iff from (p \leq q) in the second it follows that this
relation also holds in the first one? So the modified coproduct you
describe would be the least element in this partial order of generalized
coproducts. Furthermore, this modified coproduct, its dual and the
normal coproduct would be the only ones that could actually made into a
functor from Pos*Pos to Pos, since these are the only ones describable
through of the generalized coproducts describable through a universal
property: mapping each two partial orders to the lowest element in the
partial order of the lattice in the first case, to the highest one in
the second, and to the coproduct in the third. Of course it all depends
on what you consider nice, but this is the nicest I can see, at least.

I hope this is at least halfway correct (although I think it should be,
since I spent some time recently contemplating a similar structure I
found in a pet project of mine) and comprehensibly formulated. English
is not my first language and describing mathematical structures in
writing without the use of latex is a bit hard for me. I can work it out
a bit more formal in a latex file if you want.

Hope I could help,

Alex

On 13.08.2017 21:55, Dana Scott wrote:
> The category of posets (= partially ordered sets) and monotone
> maps is often used as an easy example -- different from the category
> of sets -- that has products, coproducts, and is cartesian closed
> but not a topos.
>
> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?
>



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