nLab excisive (∞,1)-functor

Idea

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

Moreover, if here $\mathcal{D}$ is instead ∞Grpd (i.e. homotopy types), or more generally any (∞,1)-topos $\mathbf{H}$, 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 such restriction are equivalent to parameterized spectra, hence form the tangent (∞,1)-topos $T \mathbf{H}$. (See also here at n-excisive functor and at Joyal locus.)

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

Definition

Definition

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

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

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

Examples

Spectrum objects

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.

Proposition

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

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

(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 $E$ be a reduced excisive functor. For each $n \in \mathbb{N}$, write $S^n \in \infty Grpd_{fin}^{\ast/}$ for the n-sphere and write $E_{n} \coloneqq E(S^n)$. We have the homotopy pushout squares

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

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

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

This makes $E_\bullet$ 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 $X$ is then $\Omega^\infty(E \wedge X)$ (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 $E$ via taking stable homotopy groups of smash products: $X \mapsto E_\bullet(X) \coloneqq \pi_\bullet(E \wedge X)$. 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 $X \mapsto \Omega^\infty(E\wedge X)$.

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 $\mathcal{C} \to \mathcal{D}$ for $\mathcal{C} = \infty Grpd_{fin}^{\ast/}$ and $\mathcal{D} = \mathbf{H}$, hence on

$$Exc(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq T \mathbf{H} \hookrightarrow \mathbf{H}[X_\ast] \,.$$

Definition

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

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

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

Reflection and excisive approximation

Definition

Write

$$T_1 \colon \mathbf{H}[X_\ast] \longrightarrow \mathbf{H}[X_\ast]$$

for the functor given by

$$E \mapsto \Omega_{E(\ast)} E(\Sigma(-)) \,.$$

Write

$$P_1 \coloneqq \underset{\longrightarrow}{\lim}_{n} T_1^{n}$$

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 $\mathbf{H}[X_*]$ is a reflective sub-(∞,1)-category with reflector given by $P_1$ from def. :

$$T \mathbf{H} \stackrel{\overset{P_1}{\longleftarrow}}{\hookrightarrow} \mathbf{H}[X_\ast]$$

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

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

$$P_1 F \simeq \underset{\longrightarrow_{\mathrlap{k}}}{\lim} ( \Omega^k \circ F \circ \Sigma^k)$$

(where $\Omega$ and $\Sigma$ denote looping and suspension in $\mathcal{D}$ and in $\mathcal{C}$, 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 $\infty Grpd_{fin}^{\ast/}$, 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-Biederman-Finster-Joyal.

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

Definition

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

$$X^{S_\ast} \in \mathcal{C}$$

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

$$(\infty Grpd_{fin}^{\ast/})^\op \longrightarrow \mathcal{C}$$

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

Proposition

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

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

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

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

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

Proof

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

$$R(K)^S \simeq R(S \cdot K) \,.$$

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

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

Hence

$$X_\ast^S \simeq R(S) \,.$$

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

$$\begin{aligned} Hom( \Sigma \Omega R(K), E ) &\simeq \Omega_{E(\ast)} Hom(\Omega R(K),E) \\ & \simeq \Omega_{E(\ast)} Hom( R(\Sigma K), E ) \\ & \simeq \Omega_{E(\ast)} E(\Sigma K) \end{aligned} \,.$$

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

$$\begin{aligned} Hom(\Sigma \Omega R(K) \to R(K), E) & \simeq ( E(K) \longrightarrow \Omega(\Sigma K) ) \\ & = (E \to T_1 E)(K) \end{aligned} \,,$$

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 $E$ for which the morphism

$$E \longrightarrow T_1 E$$

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

Related concepts

• exact (∞,1)-functor

• n-excisive (∞,1)-functor

• model structure for excisive functors

• spectrum object

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

• Thomas Goodwillie, Calculus III: Taylor Series, Geom. Topol. 7(2003) 645-711 (arXiv:math/0310481)

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

• Jacob Lurie, section 6.1.1 of Higher Algebra

Review includes:

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

On excisive 1-functors for genuine G-spectra (i.e. in proper equivariant stable homotopy theory):

• Dieter Degrijse, Markus Hausmann, Wolfgang Lück, Irakli Patchkoria, Stefan Schwede, Section 3.1 of: Proper equivariant stable homotopy theory, Memoirs of the AMS [arXiv:1908.00779]

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

• Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, Goodwillie’s Calculus of Functors and Higher Topos Theory (arXiv:1703.09632)