Re: colimits of polynomial functors

["Paul Taylor" <pt11@PaulTaylor.EU>]

Ondrej Rypacek asked,

> Does the category of (dependent) polynomial functors and strong
> natural transformation have all/some colimits ?
> In general, what is known about them ?

I studied polynomial functors under the name of "stable" functors
in categorical domain theory between approx 1987 and 1993:
     www.PaulTaylor.EU/stable/

I am guessing that, by "strong" natural transformations you mean
those for which the naturality squares are pullbacks, which I
called "cartesian".

I studied cartesian closed 2-categories whose 1- and 2-cells
are stable functors and cartesian natural transformations.

Yes, there are interesting colimits here, although they are multi-
or poly-valued.  Multi-colimits had been introduced by Yves Diers.
I don't remember who introduced poly-colimits, but these are ones
indexed by groupoids instead of sets.

"Quantitative Domains, Groupoids and Linear Logic" was probably
my most readable paper on this topic.

This kind of domain theory was begun by Gerard Berry and popularised
by Jean-Yves Girard.   In its categorical form, Francois Lamarche
also did work of the same kind as mine, except with a weaker
notion of "cartesian" that had been introduced by Andre Joyal.

Paul Taylor





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