nLab
n-excisive (∞,1)-functor

Contents

Contents

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.

(e.g. Lurie, def. 6.1.1.2 with prop. 6.1.1.15)

nn-Excisive functors

Definition

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

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

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

Remark

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

Properties

nn-Excisive approximation and reflection

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

Definition

For nn \in \mathbb{N}

Exc n(𝒞,𝒟)Func(𝒞,𝒟) 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 nn-excisive.

Proposition

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

P n:Func(𝒞,𝒟)Exc n(𝒞,𝒟) 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 nP_n is in (Lurie, constrution 6.1.1.27).

For n=1n = 1 this reflection is spectrification.

Corollary

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

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

is an (∞,1)-topos. (For n>1n \gt 1 this is in general not a hypercomplete (∞,1)-topos, even if H\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(𝒞,H)PSh(𝒞 op,H)Exc^n(\mathcal{C}, \mathbf{H}) \hookrightarrow PSh(\mathcal{C}^{op}, \mathbf{H}) is the Weiss topology on 𝒞 op\mathcal{C}^{op}.

Remark

As nn ranges, the tower of nn-excisive approximations of an (,1)(\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

P n+1FP nFP 1FP 0F. \cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F \,.

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

Definition

In the situation of def. , the functors FF for which P n1F*P_{n-1}F \simeq \ast (hence the P n1P_{n-1} anti-modal types) are called nn-reduced (∞,1)-functors.

(e.g. Lurie, def. 6.1.2.1)

Homogeneous pieces

A polynomial \infty-functor of degree kk — that is, a kk-excisive functor which is not nn-excisive for any n<kn\lt k — is a homogeneous polynomial if its approximation by an (k1)(k-1) degree polynomial is trivial.

(…)

Proposition

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

Then for all natural numbers n1n \geq 1 composition with the forgetful functor 𝒞 */𝒞\mathcal{C}^{\ast/} \to \mathcal{C} induces an equivalence of (∞,1)-categories

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

(Lurie, prop. 6.1.2.11)

Examples

Goodwillie nn-jets

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

Example

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

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

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

Hence one might refer to the tower of toposes

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

with

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

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

see (Lurie, def. 1.4.2.8 and around p. 823)

References

The notion of nn-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 nn-excisive functors was then studied in

See also

A discussion in the general abstract context of (∞,1)-category theory is in

A model structure for n-excisive functors is given in

Relation to Mackey functors is discussed in

Last revised on December 21, 2017 at 05:19:32. See the history of this page for a list of all contributions to it.