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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Dear Dana, Let (P_i | i in I) be a family of posets and < a well-ordering of I. The <-lexicographic sum of (P_i | i in I) is given by Sum_{i in I} P_i with (i,x) <= (j,y) if i=j and x<=y or i<j. It may be viewed as a representing object as follows. A cocone (f_i | i in I) in Poset from (P_i | i in I) to a poset V is "<-lexicographic" when for all i < j and x in P_i and y in P_j we have f_i(x) <= f_i(y). (*) So we obtain a right Poset-module, i.e. functor Poset --> Set sending V to the set of <-lexicographic cocones from (P_i | i in I) to V. The <-lexicographic sum is a representing object for this functor. Therefore, for fixed (I,<), it extends uniquely to a functor Poset^I --> Poset making the representation natural. However, property (*) is not "categorical" in the sense of making sense in an arbitrary category. So this probably doesn't answer your question. Paul On 13/08/17 20:55, 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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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?

Hi Dana, unless I misunderstand the question, (<) is the join operation, which makes sense more generally for categories, and more generally for simplicial sets, or augmented simplicial sets. Here it is simply the cocontinuous extension (in each variable) of ordinal sum (i.e. the Day convolution tensor product of ordinal sum). (It plays an crucial role in the development of higher category theory, thanks to the discovery by Andr?? Joyal that slice and coslice can be defined as right adjoints to join with a fixed object. (These are generalised slices and coslices, with the classical notions corresponding to the cases of join with a point.) This is the construction that allows for the definition of limits and colimits in infinity-categories, and hence the starting point for generalising category theory from categories to infinity- categories.) [A. Joyal: Quasi-categories and Kan complexes, JPAA 2002] Cheers, Joachim. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]