nLab
excisive (∞,1)-functor

Contents

Contents

Idea

An excisive (∞,1)-functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} is one which sends homotopy pushout squares to homotopy pullback squares. If 𝒞\mathcal{C} is pointed finite homotopy types and 𝒟\mathcal{D} is spectra, then this condition is the axiom of excision in generalized homotopy, whence the name.

Moreover, if here 𝒟\mathcal{D} is instead ∞Grpd (i.e. homotopy types), or more generally any (∞,1)-topos H\mathbf{H}, then excisive functors that send the point to the point (up to equivalence) are still equivalent to spectra (spectrum objects) – essentially by the Brown representability theorem – and those without restriction are equivalent to parameterized spectra, hence form the tangent (∞,1)-topos THT \mathbf{H}.

As such, excisive functors are the lowest nontrivial stage in the Goodwillie-Taylor tower that approximates the classifying (∞,1)-topos H[X ]\mathbf{H}[X_\bullet] for pointed objects. The higher stages of this tower are given by the n-excisive (∞,1)-functors.

Definition

Definition

An (∞,1)-functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} out of an (∞,1)-category with finite (∞,1)-colimits is excisive if it takes (∞,1)-pushout squares in 𝒞\mathcal{C} to (∞,1)-pullback squares 𝒟\mathcal{D}.

This is the n=1n=1 case of the concept of n-excisive (∞,1)-functor.

(e.g. HigherAlg, def. 1.4.2.1.)

Examples

Spectrum objects

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.

Proposition

Let 𝒞\mathcal{C} be an (∞,1)-category with finite (∞,1)-limits. Then spectrum objects in 𝒞\mathcal{C} are equivalently reduced excisive (∞,1)-functor of the form

Grpd fin */𝒞. \infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C} \,.

(Sometimes, e.g. in Lurie, def. 1.4.2.8, this is taken as the very definition of spectrum objects).

A proof of prop. passing through model category presentations for excisive \infty-functors and of the Bousfield-Friedlander model structure for sequential spectra is due to (Lydakis 98), see at model structure for excisive functors at Relation to BF-model structure on sequential spectra.

The idea of the equivalence is as follows. Let EE be a reduced excisive functor. For each nn \in \mathbb{N}, write S nGrpd fin */S^n \in \infty Grpd_{fin}^{\ast/} for the n-sphere and write E nE(S n)E_{n} \coloneqq E(S^n). We have the homotopy pushout squares

S n * * S n+1 \array{ S^n &\longrightarrow& \ast \\ \downarrow & & \downarrow \\ \ast &\longrightarrow& S^{n+1} }

and since EE sends them to homotopy pullbacks with the point going to the point, this gives equivalences

E nΩE n+1. E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1} \,.

This makes E E_\bullet have the structure of an Omega spectrum. The idea then is that as such it represents a generalized homology theory and the value of the excisive functor on any finite homotopy type XX is then Ω (EX)\Omega^\infty(E \wedge X) (see Lurie, remark 1.4.3.3).

Remark

The traditional definition of a generalized homology theory is as a functor on (finite) homotopy types with values in graded abelian groups. The Brown representability theorem says that these all arise from spectra EE via taking stable homotopy groups of smash products: XE (X)π (EX)X \mapsto E_\bullet(X) \coloneqq \pi_\bullet(E \wedge X). But due to the existence of phantom maps, this does not quite yield an equivalence between spectra and generalized homology theories.

In view of this, the proof of prop. may be thought of saying that this mismatch is fixed by refining homotopy groups by full homotopy types XΩ (EX)X \mapsto \Omega^\infty(E\wedge X).

Notice also that spectra realized as excisive functors this way are in the spirit of coordinate-free spectra.

Properties

For the moment we focus on properties of the case of excisive functors 𝒞𝒟\mathcal{C} \to \mathcal{D} for 𝒞=Grpd fin */\mathcal{C} = \infty Grpd_{fin}^{\ast/} and 𝒟=H\mathcal{D} = \mathbf{H}, hence on

Exc(Grpd fin */,H)THH[X *]. Exc(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq T \mathbf{H} \hookrightarrow \mathbf{H}[X_\ast] \,.
Definition

Write Grpd fin\infty Grpd_{fin} is the (∞,1)-category of finite homotopy types. For H\mathbf{H} a given base (∞,1)-topos, write

H[X *] [Grpd fin */,H] PSh((Grpd fin */) op,H) \begin{aligned} \mathbf{H}[X_\ast] & \simeq [\infty Grpd_{fin}^{\ast/}, \mathbf{H}] \\ & \simeq PSh((\infty Grpd_{fin}^{\ast/})^{op}, \mathbf{H}) \end{aligned}

for the classifying (∞,1)-topos (over H\mathbf{H}) for pointed objects.

Reflection and excisive approximation

Definition

Write

T 1:H[X *]H[X *] T_1 \colon \mathbf{H}[X_\ast] \longrightarrow \mathbf{H}[X_\ast]

for the functor given by

EΩ E(*)E(Σ()). E \mapsto \Omega_{E(\ast)} E(\Sigma(-)) \,.

Write

P 1lim nT 1 n P_1 \coloneqq \underset{\longrightarrow}{\lim}_{n} T_1^{n}

for the homotopy colimit of the iterations of this functor, with respect to the canonical comparison map.

Unwinding the definition and using that suspension is equivalently the join with the 0-sphere, this is indeed the functor of the same name in (Goodwillie 91, p. 657 (13 of 67)).

Theorem

The inclusion of excisive functors into H[X *]\mathbf{H}[X_*] is a reflective sub-(∞,1)-category with reflector given by P 1P_1 from def. :

THP 1H[X *] T \mathbf{H} \stackrel{\overset{P_1}{\longleftarrow}}{\hookrightarrow} \mathbf{H}[X_\ast]

This is due (in the generality of n-excisive functor – n-Excisive Approximation and reflection) to (Goodwillie 91, theorem 1.8). See also (Lurie, theorem 6.1.1.10, construction 6.1.1.27).

Proposition

Let 𝒞\mathcal{C} have finite colimits and a terminal object and let 𝒟\mathcal{D} be differentiable.

The excisive approximation of a reduced functor F:𝒞𝒟F \colon \mathcal{C} \to \mathcal{D} is the (infinity,1)-colimit

P 1Flim k(Ω kFΣ k) P_1 F \simeq \underset{\longrightarrow_{\mathrlap{k}}}{\lim} ( \Omega^k \circ F \circ \Sigma^k)

(where Ω\Omega and Σ\Sigma denote looping and suspension in 𝒟\mathcal{D} and in 𝒞\mathcal{C}, respectively).

(Lurie, example 6.1.1.28)

Remark

Under the equivalence to sequential spectra in prop. , the formula in prop. is the standard formula for spectrification of prespectra.

Characterization via a generic stable object

We discuss a characterization of excisive functors on Grpd fin */\infty Grpd_{fin}^{\ast/}, hence of parameterized spectra, as the result of forcing a generic pointed object to become a stable homotopy type. This general perspective is being highlighted by Anel-Finster-Joyal.

For a slick formulation, we use a generalization of powering to pointed powers:

Definition

For XX an object in an (∞,1)-category 𝒞\mathcal{C} with finite (∞,1)-limits, and for S *Grpd fin */S_\ast \in \infty Grpd_{fin}^{\ast/} a pointed finite ∞-groupoid, then the pointed power

X S *𝒞 X^{S_\ast} \in \mathcal{C}

is the object which is the image of SS under the essentially unique (∞,1)-functor

(Grpd fin */) op𝒞 (\infty Grpd_{fin}^{\ast/})^\op \longrightarrow \mathcal{C}

which preserves finite (∞,1)-limits and sends S 0*S^0 \leftarrow \ast to X **X_\ast \to \ast.

Proposition

Excisive functors Grpd fin */H\infty Grpd_{fin}^{\ast/} \longrightarrow \mathbf{H}, def. , are the localization of H[X *]\mathbf{H}[X_\ast], def. , at the set of morphisms

{ΣΩ(X * S *)X * S *} SGrpd fin */, \left\{ \Sigma \Omega (X_\ast^{S_\ast}) \longrightarrow X_\ast^{S_\ast} \right\}_{S \in \infty Grpd_{fin}^{\ast/}} \,,

(where X * S *X_{\ast}^{S_\ast} is the pointed power, def. , of the generic pointed object X *H[X *]X_\ast \in \mathbf{H}[X_\ast]):

THH[X *][(ΣΩX * X * ) 1]. T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ] \,.

In other words, the parameterized spectra are those objects in H[X *]\mathbf{H}[X_\ast] which regard each finite pointed power of the generic pointed object X *X_\ast as a stable homotopy type.

Proof

For KGrpd fin */K \in \infty Grpd_{fin}^{\ast/}, write R(K)(Grpd fin */) opH[X *]R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast] for its formal dual under (∞,1)-Yoneda embedding. Since the (∞,1)-Yoneda embedding preserves (∞,1)-limits, we have

R(K) SR(SK). R(K)^S \simeq R(S \cdot K) \,.

Observe that the generic pointed object in H[X *]\mathbf{H}[X_\ast] is that represented by the 0-sphere:

X *=R(S 0). X_\ast = R(S^0) \,.

Hence

X * SR(S). X_\ast^S \simeq R(S) \,.

Now using the (∞,1)-Yoneda lemma we have for each EH[X *]E \in \mathbf{H}[X_\ast] that

Hom(ΣΩR(K),E) Ω E(*)Hom(ΩR(K),E) Ω E(*)Hom(R(ΣK),E) Ω E(*)E(ΣK). \begin{aligned} Hom( \Sigma \Omega R(K), E ) &\simeq \Omega_{E(\ast)} Hom(\Omega R(K),E) \\ & \simeq \Omega_{E(\ast)} Hom( R(\Sigma K), E ) \\ & \simeq \Omega_{E(\ast)} E(\Sigma K) \end{aligned} \,.

Hence for all KGrpd fin */K \in \infty Grpd_{fin}^{\ast/}

Hom(ΣΩR(K)R(K),E) (E(K)Ω(ΣK)) =(ET 1E)(K), \begin{aligned} Hom(\Sigma \Omega R(K) \to R(K), E) & \simeq ( E(K) \longrightarrow \Omega(\Sigma K) ) \\ & = (E \to T_1 E)(K) \end{aligned} \,,

where in the last line we observe that the expression is that for the comparison map in def. .

This means that the local objects are precisely those EE for which the morphism

ET 1E E \longrightarrow T_1 E

from def. is an equivalence. With this the statement follows from theorem .

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 nn-excisive functors was then studied in

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

Review includes

  • Yonatan Harpaz, section 5 of Introduction to stable \infty-categories, October 2013 (pdf)

A model structure for excisive functors was given in

  • Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)

Discussion in terms of stable homotopy types is due to

Last revised on June 2, 2019 at 14:55:03. See the history of this page for a list of all contributions to it.