# nLab polynomial (∞,1)-functor

Polynomial -functors

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.

# Polynomial $(\infty,1)$-functors

## 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

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

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)).