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/ ]