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From: Paul Blain Levy <P.B.Levy@cs.bham.ac.uk>
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Subject: Re: An elementary question
Date: Mon, 14 Aug 2017 09:00:36 +0100
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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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