Re: An elementary question

[Patrik Eklund <peklund@cs.umu.se>]

What would be the practical applications of that construction?

If P and Q are two-pointed, truth in one is falser than false in the
other. If P and Q are powersets, the full set in one is more empty than
the empty in the other.

Even the mathematical justfication of that construction is a bit far
fetched, isn't it?

However, we might have those posets as economists and mathematicians.
The best economist is worse than the worst mathematician, or the best
mathematician is worse than the worst economist. Such attitudes do exist
but they are not very practical, are they?

The general intuition, however, if my intuition about what the general
intuition is in this situation is correct or at least ordered, is
interesting when going towards ordering categorical objects based on
structure the objects respectively embrace. Monoidal categories
involving that tensor are interesting, and we already have a fair
understanding about what they can do for us in many application areas,
so introducing non-commutativity via modified coproducts sounds like
something that might be already in the making. I wouldn't be surprised
at all if that indeed is the case.

Note also that the objects themselves are not the whole story but that
"orderal" or "posetal" category may turn out to be a most interesting
underlying category for many applications involving applications. We
indeed already see that in the case of monoidal categories.

Looking forward to formulations of monoidal-posetal categories!

Best,

Patrik



On 2017-08-13 22:55, 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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