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Stream: theory: category theory

Topic: Ran of polynomials

Jules Hedges (Aug 20 2025 at 16:21):

The category of polynomial endofunctors (over Set) is closed under left Kan extensions - I only learned this recently but I think it's well known to people who know about polynomials. I think it's not closed under right Kan extensions, but I don't really trust my intuition, since I also recently learned that the category of polynomials is cartesian closed (by an extremely not-obvious construction) after believing for years that it wasn't.

Are polynomials closed under Ran? If not, is there a nice description of the smallest category that contains all polynomials and is closed under Ran? It would be very nice if all endofunctors arise this way, but after thinking about it for a bit I don't think that's true either

Jules Hedges (Aug 20 2025 at 16:24):

(Apart from intrinsic motivation this question also has some relevance to programming languages, since any language with algebraic datatypes and polymorphic types can construct these things)

Kevin Carlson (Aug 20 2025 at 18:13):

The coequalizer of the two projections $y^2\rightrightarrows y$ in Poly is terminal, but plugging in $0$ to this coequalizer diagram gives $0\rightrightarrows 0\to 1$. So composition is not cocontinuous on the left, thus not left closed and Poly isn’t closed under Ran. I don’t know about your other question right away.

Kevin Carlson (Aug 20 2025 at 18:18):

In retrospect, you could make this argument a smidge quicker by just calculating that the right Kan extension of a polynomial $p$ along the empty functor sends $0$ to $p(0)$ and everything else to $1$, which is not polynomial unless $p(0)=1.$