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Stream: theory: category theory

Topic: are containers polynomial functors?

Li-yao Xia (Sep 15 2021 at 03:38):

This paper seems to be spelling it out but I can't see how Application 1 (containers) is an instance of Definition 2 (polynomials): https://pages.lip6.fr/Pierre-Evariste.Dagand/stuffs/report-2011-catorn/paper.pdf

Kenji Maillard (Sep 15 2021 at 08:08):

Given an indexed container $(S : J \to Set, P : S\,j\to Set, in : P\,sh \to J)$ you obtain a polynomial
$I \leftarrow \Sigma (j : J) \Sigma (sh : S\,j)~P\,sh \to \Sigma (j : J)~S\,j\to J$ in any (LCCC) model of your type theory, where

• the first arrow maps $(j, sh, p)$ to $in\,p : I$,
• the second arrow projects out the two first components of $(j, sh, p)$ to $(j, sh) : \Sigma (j : J)~S\,j$, and
• the third arrow projects out the first component of $(j, sh)$ to $j : J$.
  Is that the description you are asking for ?

Li-yao Xia (Sep 15 2021 at 11:39):

Exactly what I was looking for, thanks!