Re: colimits of polynomial functors

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

Hi Ondrej,

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

At the risk of being off the point, I think that the colimits
that exist might not have been studied much because they are often
not the 'right' ones, in a sense.  As an example, the polynomial
functor Set -> Set, X \mapsto X^2 (represented by 1 <- 2 -> 1 -> 1)
has two automorphisms (the identity and the twist), and if I am not
mistaken the identity functor X \mapsto X is the coequaliser of those
two in the category of polynomial functors and their strong natural
transformations (just because 1 is the equaliser of the two set auts
2 -> 2).  The functor that 'ought' to be the coequaliser is of course
X \mapsto X^2/2, which is not polynomial.  (For example it does not
preserve pullbacks.)

(I should add that I understand the question as concerning
polynomial functors and those natural transformations compatible
with the canonical tensorial strengths.  If instead, according to
Paul Taylor's interpretation of the question, only cartesian
natural transformations are allowed, then it is easy to see
that the above pair of (cartesian) natural transformations
does not have a coequaliser.)

Cheers,
Joachim.

PS: allow me to advertise a reference:
[Gambino-Kock, Polynomial functors and polynomial monads, arXiv 2009].
It does not have anything about colimits, but it does say a lot about
strong natural transformations, and in particular characterise them
in diagrammatic terms.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]