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added to polynomial monad the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.
I added some discussion of “cartesian polynomial monads” to polynomial monad, and their relationship to clubs and object classifiers. I am currently finding this perspective very helpful; for some reason polynomial functors feel like a much more natural thing to me than clubs, so it is nice that they are almost the same thing.
All the references at polynomial monad define a polynomial monad to be one whose underlying endofunctor is a polynomial functor and whose unit and multiplication are cartesian natural transformations. I am inclined to alter our page polynomial monad to conform to this terminology, unless I hear objections.
If there are interesting examples of non-cartesian polynomial monads, then the terminology on our page is a useful generalization. (But there should still be a note added that most references require all polynomial monads to be cartesian.) If not, then sure, change it.
I don’t know of any such examples. But even if there were such examples, for the terminology to be a useful generalization I would argue that there ought also to be interesting theorems about non-cartesian polynomial monads, and I don’t know any of those either.
Ok, I made the change.
Added
An extension of type theory with a universe of associative and unital polynomial monads to define opetopic types in order to encode fully coherent algebraic structures is in
- {#AlliouxFinsterSozeau20} Antoine Allioux, Eric Finster, Matthieu Sozeau, Types are internal infinity-groupoids (PDF)
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