This page is about a categorification of the notion of polynomial functor. For the notion of “polynomial -functor” used in Goodwillie calculus, see n-excisive (∞,1)-functor.
A polynomial -functor is a categorification of the notion of polynomial functor.
Recall that a map of spaces induces an adjoint triple
where is the base change functor.
A polynomial -functor is a functor equivalent to a functor of the form , where
are maps of spaces.
Polynomial functors are closed under compositions (GHK, Theorem 2.1.8).
A functor is polynomial if and only if it is accessible and preserves weakly contractible limits, the latter referring to limits indexed by categories whose nerve is a weakly contractible simplicial set, see (GHK, Theorem 2.2.3(ii)).
Recall that an -functor is a local right adjoint functor if for any object the induced functor
is a right adjoint functor.
A functor is polynomial if and only if it is a local right adjoint functor, see (GHK, Theorem 2.2.3(iii)).
David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, (arXiv:1712.06469)
Last revised on March 8, 2021 at 20:42:02. See the history of this page for a list of all contributions to it.