nLab polynomial (∞,1)-functor

This page is about a categorification of the notion of polynomial functor. For the notion of “polynomial $\infty$-functor” used in Goodwillie calculus, see n-excisive (∞,1)-functor.

Idea

A polynomial $(\infty,1)$-functor is a categorification of the notion of polynomial functor.

Definition

Recall that a map of spaces $f\colon I\to J$ induces an adjoint triple

$$f_!\dashv f^*\dashv f_*,$$

where $f^*\colon S/J\to S/I$ is the base change functor.

A polynomial $(\infty,1)$-functor is a functor $S/I\to S/J$ equivalent to a functor of the form $t_! p_* s^*$, where

$$I \leftarrow E \to B\to J$$

are maps of spaces.

Polynomial functors are closed under compositions (GHK, Theorem 2.1.8).

A functor $F\colon S/I\to S/J$ 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 $(\infty,1)$-functor $F\colon C\to D$ is a local right adjoint functor if for any object $X\in C$ the induced functor

$$C/X\to D/F(X)$$

is a right adjoint functor.

A functor $F\colon S/I\to S/J$ is polynomial if and only if it is a local right adjoint functor, see (GHK, Theorem 2.2.3(iii)).

Related pages

• (∞,1)-operad

References

David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, (arXiv:1712.06469)