From: Paul Blain Levy <P.B.Levy@cs.bham.ac.uk>
To: Dana Scott <scott@cs.cmu.edu>, categories@mta.ca
Subject: Re: An elementary question
Date: Mon, 14 Aug 2017 09:00:36 +0100 [thread overview]
Message-ID: <E1dhK2m-00059i-HU@mlist.mta.ca> (raw)
In-Reply-To: <E1dh2U9-0000c9-3y@mlist.mta.ca>
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?
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2017-08-14 8:00 UTC|newest]
Thread overview: 11+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-08-13 19:55 Dana Scott
2017-08-14 0:15 ` alex
2017-08-14 4:42 ` Patrik Eklund
2017-08-14 18:43 ` Mike Stay
2017-08-15 5:57 ` Vaughan Pratt
[not found] ` <fa2a57444ecc56bfc61165f2263d42e5@cs.umu.se>
2017-08-15 14:21 ` Mike Stay
2017-08-14 8:00 ` Paul Blain Levy [this message]
2017-08-14 18:51 ` Robin Cockett
[not found] ` <CAKQgqTb4f=+Y=SuCu26AKg3Wd02friY_6z9FhukXiSnoapLZRQ@mail.gmail.com>
2017-08-15 4:50 ` Patrik Eklund
2017-08-15 21:49 ` Joachim Kock
2017-08-17 2:02 ` Branko Nikolić
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