nLab
analytic (∞,1)-functor

Contents

Contents

Idea

In the context of Goodwillie calculus an (∞,1)-functor is called analytic if it behaves in analogy with an analytic function in that it its Goodwillie-Taylor tower converges to it.

Definition

(Co-)Cartesian cubical diagrams

Let 𝒞\mathcal{C} be an (∞,1)-category with finite (∞,1)-colimits.

Definition

An nn-cube in 𝒞\mathcal{C}, hence an (∞,1)-functor n𝒞\Box^n \longrightarrow \mathcal{C}, is called strongly homotopy co-cartesian or just strongly co-cartesian, if all its 2-dimensional square faces are homotopy pushout diagrams in 𝒞\mathcal{C}.

Definition

An nn-cube in 𝒟\mathcal{D}, hence an (∞,1)-functor n𝒟\Box^n \longrightarrow \mathcal{D}, is called homotopy cartesian or just cartesian, if its “first” object exhibits a homotopy limit-cone over the remaining objects.

Analytic functors

Definition

An (∞,1)-functor F:𝒞𝒟F \colon \mathcal{C} \to \mathcal{D} is stably nn-excisive with constants cc and κ\kappa_ – or satisfies “condition E n(c,κ)E_n(c,\kappa) – if for every strongly co-Cartesian (n)+1(n)+1-cube XX in 𝒞\mathcal{C}, def. , such that X()X(s)X(\emptyset) \to X(s) is k sk_s-connective for k sκk_s \geq \kappa for all s{1,,n+1}s\in \{1,\cdots, n+1\}, then F(X)F(X) is an (n+1)(n+1)-cube in 𝒟\mathcal{D} such that the comparison map

F()limKsubcubeofF(X)KF(K) F(\emptyset) \longrightarrow \underset{{K\,subcube\,of\,F(X)}\atop {K \neq \emptyset}}{\lim} F(K)

(to the indicated homotopy limit) is (c+ sk s)(-c + \sum_s k_s)-connective.

The functor FF is called ρ\rho-analytic if there is qq such that it satisfies the condition E n(nρq,ρ+1)E_n(n\rho - q,\rho + 1) for all nn.

(e.g. Johnson 95, def. 1.1, def. 1.3)

Properties

Convergence of the Goodwillie-Taylor tower

For ρ\rho-analytic functors their Goodwillie-Taylor tower converges to them on ρ\rho-connective objects. See there.

Examples

The identity functor on homotopy types

The identity (,1)(\infty,1)-functor on ∞Grpd is 1-analytic, def. . For n=2n = 2 this is the statement of the Blakers-Massey theorem, for n>2n \gt 2 this is the statment of the higher cubical BM-theorems.

(see e.g. Munson-Volic 15, example 10.1.18)

References

The concept is due to

  • Tom Goodwillie, Calculus II: Analytic functors, K-Theory 01/1991; 5(4):295-332. DOI: 10.1007/BF00535644

Review includes

  • Tom Goodwillie, section 3 of The differential calculus of homotopy functors (pdf)

  • Brenda Johnson, The derivatives of homotopy theory, Transactions of the AMS, Volume 347, 1995 (pdf)

A textbook account is in

Last revised on January 5, 2016 at 04:37:35. See the history of this page for a list of all contributions to it.