Contents

Contents

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.

$n$-Excisive functors

Definition

An (∞,1)-functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is $n$-excisive for $n \in \mathbb{N}$ (or polynomial of degree at most $n$) if whenever $X$ is a strongly cocartesian $(n + 1)$-cube in $\mathcal{C}$, def. , then $F(X)$ is a cartesian cube in $\mathcal{D}$, def. .

A 1-excisive (∞,1)-functor is often just called excisive (∞,1)-functor for short.

An $(\infty,1)$-functor which is $n$-excisive for some $n \in \mathbb{N}$ is also called a polynomial (∞,1)-functor (not to be confused with other concepts having the same name). It has degree $k$ when the smallest value of $n$ for which it is $n$-excisive is $k$.

Remark

This notion is comparable to how a polynomial of degree at most $n$ is determined by its values on $n + 1$ distinct points.

Properties

$n$-Excisive approximation and reflection

Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and a terminal object, and let $\mathcal{D}$ be a Goodwillie-differentiable (∞,1)-category.

Definition

For $n \in \mathbb{N}$

$Exc^n(\mathcal{C}, \mathcal{D}) \hookrightarrow Func(\mathcal{C}, \mathcal{D})$

for the full sub-(∞,1)-category of the (∞,1)-functor (∞,1)-category on those (∞,1)-functors which are $n$-excisive.

Proposition

The inclusion of def. is lex reflective, hence the inclusion functor has a left adjoint (∞,1)-functor

$P_n \;\colon\; Func(\mathcal{C}, \mathcal{D}) \longrightarrow Exc^n(\mathcal{C}, \mathcal{D})$

which moreover is left exact (preserves finite (∞,1)-limits).

This is essentially the statement of (Goodwillie 03, theorem 1.8). In the above form it appears explicitly as (Lurie, theorem 6.1.1.10). The construction of the reflector $P_n$ is in (Lurie, constrution 6.1.1.27).

For $n = 1$ this reflection is spectrification.

Corollary

For $\mathcal{D} = \mathbf{H}$ an (∞,1)-topos, then for all $n \in \mathbb{N}$ we have that

$Exc^n(\mathcal{C}, \mathbf{H}) \in (\infty,1)Topos$

is an (∞,1)-topos. (For $n \gt 1$ this is in general not a hypercomplete (∞,1)-topos, even if $\mathbf{H}$ is.)

This observation is due to Charles Rezk. It appears as (Lurie, remark 6.1.1.11).

Remark

A site of definition of $Exc^n(\mathcal{C}, \mathbf{H}) \hookrightarrow PSh(\mathcal{C}^{op}, \mathbf{H})$ is the Weiss topology on $\mathcal{C}^{op}$.

Remark

As $n$ ranges, the tower of $n$-excisive approximations of an $(\infty,1)$-functor, accordding to prop. , forms a tower analogous to the the Taylor series of a smooth function. This is called the Goodwillie-Taylor tower

$\cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F \,.$

If this converges to $F$, then $F$ is analogous to an analytic function and is called an analytic (∞,1)-functor.

Definition

In the situation of def. , the functors $F$ for which $P_{n-1}F \simeq \ast$ (hence the $P_{n-1}$ anti-modal types) are called $n$-reduced (∞,1)-functors.

(e.g. Lurie, def. 6.1.2.1)

Homogeneous pieces

A polynomial $\infty$-functor of degree $k$ — that is, a $k$-excisive functor which is not $n$-excisive for any $n\lt k$ — is a homogeneous polynomial if its approximation by an $(k-1)$ degree polynomial is trivial.

(…)

Proposition

Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and with terninal object. Let $\mathcal{D}$ be a pointed Goodwillie-differentiable (∞,1)-category. Write $\mathcal{C}^{\ast/}$ for the pointed objects in $\mathcal{C}$.

Then for all natural numbers $n \geq 1$ composition with the forgetful functor $\mathcal{C}^{\ast/} \to \mathcal{C}$ induces an equivalence of (∞,1)-categories

$Homog^n(\mathcal{C},\mathcal{D}) \stackrel{\simeq}{\longrightarrow} Homog^n(\mathcal{C}^{\ast/}, \mathcal{D})$

Examples

Goodwillie $n$-jets

Write $\infty Grpd_{fin}$ for the (∞,1)-category of finite homotopy types, hence those freely generated by finite (∞,1)-colimits from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the pointed finite homotopy types.

Example

For $\mathbf{H}$ an (∞,1)-topos we have that

• $Exc^0(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq \mathbf{H}$ is the collection of constant functors, hence the original (∞,1)-topos itself;

• $Exc^1(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq T\mathbf{H}$ is the collection of parameterized spectra in $\mathbf{H}$, hence the tangent (∞,1)-topos of $\mathbf{H}$.

Hence one might refer to the tower of toposes

$\cdots \to J^n \mathbf{H} \to \cdots \to J^2 \mathbf{H} \to T \mathbf{H} \to \mathbf{H}$

with

$J^n \mathbf{H} \coloneqq Exc^n(\infty Grpd^{\ast/}, \mathbf{H})$

the tower of “Goodwillie jet (∞,1)-categories” of $\mathbf{H}$.

References

The notion of $n$-excisive functors was introduced in

• Thomas Goodwillie, Calculus II, Analytic functors, K-Theory 5 (1991/92), no. 4, 295-332

The Taylor tower formed by $n$-excisive functors was then studied in

• Charles Rezk, A streamlined proof of Goodwillie’s $n$-excisive approximation (arXiv:0812.1324)