nLab model structure for excisive functors

Idea

A model category structure for excisive functors on functors from (finite) pointed simplicial sets to pointed simplicial sets (Lydakis 98, theorem 9.2, Biedermann-Chorny-Röndings 06, section 9), hence (see here) a model structure for spectra (Lydakis 98, theorem 11.3).

Special case of a model structure for n-excisive functors.

Definition

The underlying categories

Definition

Write

• sSet for the category of simplicial sets;

• $sSet^{\ast/}$ for the category of pointed simplicial sets;

• $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the full subcategory of pointed simplicial finite sets.

Write

$$sSet^{\ast/} \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}} sSet$$

for the free-forgetful adjunction, where the left adjoint functor $(-)_+$ freely adjoins a base point.

Write

$$\wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}$$

for the smash product of pointed simplicial sets, similarly for its restriction to $sSet_{fin}^{\ast}$:

$$X \wedge Y \coloneqq cofib\left( \; \left(\, (u(X),\ast) \sqcup (\ast, u(Y)) \,\right) \longrightarrow u(X) \times u(Y) \; \right) \,.$$

This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a closed monoidal category and we write

$$[-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}$$

for the corresponding internal hom, the pointed function complex functor.

Remark

For $X,Y\in sSet^{\ast/}$, the internal hom $[X,Y] \in sSet^{\ast/}$ is the simplicial set

$$[X,Y]_n = Hom_{sSet^{\ast/}}(X \wedge \Delta[n]_+, Y)$$

regarded as pointed by the zero morphism (the one that factors through the base point), and the composition morphism

$$\circ \:\colon\; [X,Y] \wedge [Y,Z] \longrightarrow [X,Z]$$

is given by

$$\begin{aligned} \circ_n & \;\colon\; ( X \wedge \Delta[n]_+ \stackrel{f}{\longrightarrow} Y \;,\; Y \wedge \Delta[n]_+ \stackrel{g}{\longrightarrow} Z ) \\ & \mapsto ( X \wedge \Delta[n]_+ \stackrel{X \wedge (diag_{\Delta[n]})_+}{\longrightarrow} X \wedge (\Delta[n] \times \Delta[n])_+ \simeq X \wedge \Delta[n]_+ \wedge \Delta[n]_+ \stackrel{f \wedge id}{\longrightarrow} Y \wedge \Delta[n]_+ \stackrel{g}{\longrightarrow} Z ) \end{aligned} \,.$$

We regard all the categories in def. canonically as simplicially enriched categories, and in fact regard $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-enriched categories.

Remark

The smash product as an $sSet^{\ast/}$-enriched functor takes

$$\wedge \;\colon\; [X_1,Y_1] \wedge [X_2, Y_2] \longrightarrow [X_1 \wedge X_2, Y_1\wedge Y_2]$$

by

$$\begin{aligned} \wedge_n & \colon ( X_1 \wedge \Delta[n]_+ \stackrel{f_1}{\longrightarrow} Y_1 \;,\; X_2 \wedge \Delta[n]_+ \stackrel{f_2}{\longrightarrow} Y_2 ) \\ & \mapsto X_1 \wedge X_2 \wedge \Delta[n]_+ \stackrel{id \wedge diag_+}{\longrightarrow} X_1 \wedge X_2 \wedge (\Delta[n] \times \Delta[n])_+ \simeq X_1 \wedge X_2 \wedge \Delta[n]_+ \wedge \Delta[n]_+ \simeq (X_1 \wedge \Delta[n]_+) \wedge (X_2 \wedge \Delta[n]_+) \stackrel{(f_1)_n \wedge (f_2)_n}{\longrightarrow} Y_1 \wedge Y_2 \end{aligned}$$

The category that is discussed below to support a model structure for excisive functors is the $sSet^{\ast/}$-enriched functor category

$$[sSet^{\ast/}_{fin}, sSet^{\ast/}] \,.$$

(Lydakis 98, example 3.8, def. 4.4)

In order to compare this to model structures for sequential spectra we consider also the following variant.

Definition

Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial 1-sphere.

Write

$$\iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}$$

for the non-full $sSet^{\ast/}$-enriched subcategory of pointed simplicial finite sets, def. whose

• objects are the smash product powers $S^n_{std} \coloneqq (S^1_{std})^{\wedge^n}$ (the standard minimal simplicial n-spheres);

• hom-objects are

  $$[S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres} \coloneqq \left\{ \array{ \ast & for & k \lt 0 \\ im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}}) & otherwise } \right.$$

(Lydakis 98, def. 4.2)

Proposition

There is an $sSet^{\ast/}$-enriched functor

$$(-)^seq \;\colon\; [StdSpheres,sSet^{\ast/}] \longrightarrow SeqPreSpec(sSet)$$

(from the category of $sSet^{\ast/}$-enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential prespectra in sSet) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential prespectrum $X^{seq}$ with components

$$X^{seq}_n \coloneqq X(S^n_{std})$$

and with structure maps

$$\frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}$$

given by

$$S^1_{std} \stackrel{\widetilde{id}}{\longrightarrow} [S^n_{std}, S^{n+1}_{std}] \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow} [X^{seq}_n, X^{seq}_{n+1}] \,.$$

This is an $sSet^{\ast/}$ enriched equivalence of categories.

(Lydakis 98, prop. 4.3)

The model structures

Consider the $sSet^{\ast/}$-enriched functor category $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ from above.

With $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1] \in sSet^{\ast/}$ we take looping and delooping $(\Sigma \dashv \Omega)$ to mean concretely the operation on smash product and pointed exponential with this particular $S^1_{std}$:

$$(\Sigma \dashv \Omega) \coloneqq ( S^1_{std}\wedge(-) \dashv [S^1_{std},-] ) \colon sSet^{\ast/} \longrightarrow sSet^{\ast/} \,.-$$

These operations extend objectwise to $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$, where we denote them by the same symbols.

Definition

Write

$$T \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}] \longrightarrow [sSet^{\ast/}_{fin}, sSet^{\ast/}]$$

for the functor given on $X$ by

$$T X \colon K \mapsto \Omega X(\Sigma K) \,.$$

Write

$$\tau \;\colon\; id \longrightarrow T$$

for the natural transformation whose component $\tau_{X}(K) \;\colon\; X(K) \to \Omega (X(\Sigma K))$ is the $(\Sigma \dashv \Omega)$-adjunct of the canonical morphism $\Sigma X(K) \longrightarrow X(\Sigma K)$ induced from

$$X \left( \array{ K & \longrightarrow & \ast \\ \downarrow &\swArrow& \downarrow \\ \ast &\longrightarrow& \Sigma K } \right) \;\;\;\; = \;\;\;\; \array{ X(K) &&\longrightarrow&& \ast \\ \downarrow &&& \swarrow & \downarrow \\ \downarrow && \Sigma X(K) && \downarrow \\ \downarrow & \nearrow && \searrow^{\mathrlap{\tau_{X}(K)}}& \downarrow \\ \ast &&\longrightarrow && X(\Sigma K) } \,.$$

Write

$$T^\infty \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}] \longrightarrow [sSet^{\ast/}_{fin}, sSet^{\ast/}]$$

for the functor given by $X$ by the sequential colimit

$$T^\infty X \coloneqq \underset{\longrightarrow}{\lim} \left( X \stackrel{\tau_X}{\longrightarrow} T X \stackrel{T(\tau_X)}{\longrightarrow} T (T X) \stackrel{}{\longrightarrow} \simeq \right) \,.$$

Write $Fib \colon sSet^{\ast} \to sSet^{\ast}$ for any Kan fibrant replacement functor.

Say that the stabilization (spectrification) of $X$ is

$$stab(X) \coloneqq T^\infty (Fib(Lan X(Fib(-)))) \,,$$

where $Lan X \colon sSet^{\ast/} \to sSet^{\ast}$ is the left Kan extension of $X$ along the inclusion $sSet^{\ast/}_{fin} \hookrightarrow sSet^{\ast/}$.

Definition

Say that a morphism $f \colon X \to Y$ in $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ is

• a stable weak equivalence if its stabilization, def. , takes value on each $K \in sSet^{\ast/}$ in weak homotopy equivalences in $sSet^{\ast/}$;

• a stable cofibration if it has the left lifting property against those morphisms whose value on every $K \in sSet^{\ast/}$ is a Kan fibration.

(Lydakis 98, def. 9.1, def. 7.1)

Proposition

The classes of morphisms of def. , define a model category structure

$$[sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly} \,.$$

(Lydakis 98, theorem 9.2)

Properties

Relation to BF-model structure on sequential spectra

There is a Quillen equivalence between the Bousfield-Friedlander model structure on sequential spectra and the Lydakis model structure $[sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}$ from prop. .

Proposition

There is an $sSet^{\ast/}$-enriched functor

$$(-)^seq \;\colon\; [StdSpheres,sSet^{\ast/}] \longrightarrow SeqPreSpec(sSet)$$

(from the category of $sSet^{\ast/}$-enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential prespectra in sSet) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential prespectrum $X^{seq}$ with components

$$X^{seq}_n \coloneqq X(S^n_{std})$$

and with structure maps

$$\frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}$$

given by

$$S^1_{std} \stackrel{\widetilde{id}}{\longrightarrow} [S^n_{std}, S^{n+1}_{std}] \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow} [X^{seq}_n, X^{seq}_{n+1}] \,.$$

This is an $sSet^{\ast/}$ enriched equivalence of categories.

(Lydakis 98, prop. 4.3)

Proposition

The adjunction

$$(\iota_\ast \dashv \iota^\ast) \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly} \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}} [StdSpheres, sSet^{\ast/}] \underoverset{\simeq}{(-)^{seq}}{\longrightarrow} SeqPreSpec(sSet)_{BF}$$

(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. and by left Kan extension $\iota_\ast$ along $\iota$, and combined with the equivalence $(-)^{seq}$ of prop. ) is a Quillen adjunction and in fact a Quillen equivalence between the Bousfield-Friedlander model structure on sequential prespectra and Lydakis’ model structure for excisive functors, prop. .

Symmetric monoidal smash product

The excisive functors naturally carry a smash product (Lydakis 98, def. 5.1) making the model structure for 1-excisive functors a symmetric monoidal model category (Lydakis 98, section 12). Via the translation to sequential spectra of prop. this is a model for the smash product of spectra (Lydakis 98, theorem 12.5); hence it is a symmetric smash product on spectra.

A monoid with respect to this smash product (hence a ring spectrum) is equivalently a functor with smash products (“FSP”) as earlier considered in (Bökstedt 86).

Definition

Since $(sSet^{\ast/}, \wedge)$ (def. ) is a symmetric monoidal category, $[sSet^{\ast}_{fin}, sSet^{\ast/}]$ canonically becomes symmetric monoidal itself via the induced Day convolution product. We write

$$\left(\, [sSet^{\ast/}_{fin}, sSet^{\ast/}], \; \wedge_{Say} \right)$$

for this symmetric monoidal category.

Proposition

The smash product on $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ considered (Lydakis 98, def. 5.1) coincides with the Day convolution product of def. .

Proof

The Day convolution product is characterized (see this proposition) by making a natural isomorphism of the form

$$[sSet^{\ast/}_{fin}, sSet^{\ast/}](X \wedge Y, Z) \simeq [sSet^{\ast/}_{fin} \times sSet^{\ast/}_{fin}, sSet^{\ast/}](X \tilde{\wedge} Y, Z \circ \wedge)$$

where the external smash product $\tilde {\wedge}$ on the right is defined by $X \tilde{\wedge} Y \coloneqq \wedge \circ (X,Y)$. Now, (Lydakis 98, def. 5.1) sets

$$X \wedge Y \coloneqq \wedge_\ast (X \tilde{\wedge} Y)$$

where, by (Lydakis 98, prop. 3.23), $\wedge_\ast$ is the left adjoint to $\wedge^\ast(-) \coloneqq (-)\circ \wedge$. Hence the adjunction isomorphism gives the above characterization.

Proposition

Under the Quillen equivalence of prop. the symmetric monoidal Day convolution product on excisive simplicial functors (prop. ) is identified with the proper smash product of spectra realized on sequential spectra by the standard formula.

(Lydakis 98, theorem 12.5)

This implies that any incarnation of the sphere spectrum in $[sSet^{\ast}_{fin}, sSet^{\ast/}]$, possibly suitably replaced acts as the tensor unit up to stable weak equivalence. The following says that the canonical incarnation of the sphere spectrum actually is the genuine (1-categorical) tensor unit:

Definition

Write

$$\mathbb{S}_{std}\in [sSet^{\ast/}_{fin}, sSet^{\ast/}]$$

for the canonical inclusion $sSet^{\ast/}_{fin} \hookrightarrow sSet^{\ast/}$.

(the standard incarnation of the sphere spectrum in the model structure for excisive functors).

Proposition

The object $\mathbb{S}_{std}$ of def. is (up to isomorphism) the tensor unit in $([sSet^{\ast/}_{fin}, sSet^{\ast}], \wedge_{Day})$.

Proof

This is (Lydakis 98, theorem 5.9), but it is immediate with prop. , using that the tensor unit for Day convolution is the functor represented by the tensor unit in the underlying site (this proposition).

Proposition

Equipped with the Day convolution tensor product (prop. ) the Lydakis model category of prop. becomes a monoidal model category

$$\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right) \,.$$

Proof

The pushout product axiom is (Lydakis 98, theorem 12.3). Moreover (Lydakis 98, theorem 12.4), shows that tensoring with cofibrant objects preserves all stable weak equivalences, hence in particular preserves cofibrant resolution of the tensor unit.

This means that (commutative) monoids in the monoidal Lydakis model structure $\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right)$ are good models for ring spectra (E-infinity rings/A-infinity rings).

Proposition

Monoids (commutative monoids) in the Lydakis monoidal model category $\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right)$ of prop. are equivalently (symmetric) lax monoidal functors of the form

$$sSet^{\ast/}_{fin} \longrightarrow sSet^{\ast/}$$

also known as “functors with smash product” (FSPs).

(Lydakis 98, remark 5.12)

Proof

Since the tensor product is Day convolution of the smash product on $sSet^{\ast/}_{fin}$, def. , this is a special casse of a general property of Day convolution, see this proposition.

Related concepts

model structure on functors

model structure on spectra

• sequential spectrum, model structure on sequential spectra

with symmetric monoidal smash product of spectra

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

• excisive functor, model structure on excisive functors

• Model categories of diagram spectra

model structure for n-excisive functors

References

Model structure for excisive functors on simplicial sets (hence also a model structure for spectra) is discussed in:

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

A similar model structure on functors on topological spaces was given in

• William Dwyer, Localizations, in: Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II 131, Kluwer Acad. Publ. (2004) 3-28 [doi:10.1007/978-94-007-0948-5]

and also excisive functors modeled on topological spaces are the $\mathcal{W}$-spectra in

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society 82 (2001) 441-512 [pdf]

Discussion of the restriction from excisive functors to symmetric spectra includes

• Stefan Schwede, chapter I, section 7.3 of Symmetric spectra (2012)

The functors with smash products (“FSP”s) appearing in (Lydakis 98, remark 5.12) had earlier been considered in

• Marcel Bökstedt, Topological Hochschild homology. Preprint, Bielefeld, 1986

Further generalization of the model structure for excisive functor, in particular to enriched functors and to a model structure for n-excisive functors for $n \geq 1$ is discussed in

• Georg Biedermann, Boris Chorny, Oliver Röndigs, Calculus of functors and model categories, Advances in Mathematics 214 (2007) 92-115 [arXiv:math/0601221, doi:10.1016/j.aim.2006.10.009]

• Georg Biedermann, Oliver Röndigs, Calculus of functors and model categories II, Algebr. Geom. Topol. 14 (2014) 2853-2913 [arXiv:1305.2834v2, doi:10.2140/agt.2014.14.2853]