Re: An elementary question

[Joachim Kock <kock@mat.uab.cat>]

> 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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