nLab
analytic (∞,1)-functor

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

Definition

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

Definition

An $n$-cube in $\mathcal{D}$, hence an (∞,1)-functor $\Box^n \longrightarrow \mathcal{D}$, 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 $F \colon \mathcal{C} \to \mathcal{D}$ is stably $n$-excisive with constants $c$ and $\kappa$_ – or satisfies “condition $E_n(c,\kappa)$” – if for every strongly co-Cartesian $(n)+1$-cube $X$ in $\mathcal{C}$, def. , such that $X(\emptyset) \to X(s)$ is $k_s$-connective for $k_s \geq \kappa$ for all $s\in \{1,\cdots, n+1\}$, then $F(X)$ is an $(n+1)$-cube in $\mathcal{D}$ such that the comparison map

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

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

The functor $F$ is called $\rho$-analytic if there is $q$ such that it satisfies the condition $E_n(n\rho - q,\rho + 1)$ for all $n$.

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

Properties

Convergence of the Goodwillie-Taylor tower

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

Examples

The identity functor on homotopy types

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

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

Related concepts

• excisive (∞,1)-functor

• polynomial (∞,1)-functor

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

• Brian Munson, Ismar Volic, Cubical homotopy theory, Cambridge University Press, 2015 pdf