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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