Re: colimits of polynomial functors

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

I just wrote, much too quickly:

> 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).

But the last sentence is of course pure nonsense.  The equaliser of the
two set auts 2 -> 2 is 0, and the conclusion is then that the constant
polynomial functor X \mapsto 1 (represented by 1 <- 0 -> 1 -> 1) is
the coequaliser.

Sorry for the nonsense.  I don't know where I had my head.
I can hardly trust myself anymore, but if this second version is
correct, it still illustrates the point I wanted to make, namely that
the colimit is not the 'right' one.

> 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.)

Cheers,
Joachim.





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