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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 (,1)(\infty,1)-functors

Idea

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

Definition

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

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

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

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

IEBJI \leftarrow E \to B\to J

are maps of spaces.

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

A functor F:S/IS/JF\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 (,1)(\infty,1)-functor F:CDF\colon C\to D is a local right adjoint functor if for any object XCX\in C the induced functor

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

is a right adjoint functor.

A functor F:S/IS/JF\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)).

References

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

Last revised on February 19, 2020 at 08:49:37. See the history of this page for a list of all contributions to it.