Re: An elementary question

["Branko Nikolić" <branik.mg@gmail.com>]

Dear Dana,

I'm not sure if the following construction is the one you are looking
for, but it's the only categorical (in fact 2-categorical) description
I could think of, and it is related to Robin Cockett's answer.

If you view posets as categories (ncatlab.org/nlab/show/partial+order)
then P and Q can be seen as objects of the 2-category Cat of
categories, functors and natural transformations. Furthermore, instead
of functors we can look at modules (aka profunctors or distributors,
ncatlab.org/nlab/show/profunctor) and their morphisms, to get the
bicategory Mod.
The situation you described corresponds to the terminal module between
Q and P (1-cell in Mod which is a terminal object in the hom-category
Mod(Q,P)). The poset you obtain by taking the "modified coproduct" has
the universal property of being the lax colimit of that 1-cell...
A more general construction is explained here
http://maths.mq.edu.au/~street/Pow.fun.pdf

Best regards,
Branko

On 16 Aug 2017 11:50 pm, "Joachim Kock" <kock@mat.uab.cat> wrote:
>
> 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.


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