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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