nLab
model structure for excisive functors

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Stable Homotopy theory

Contents

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

Write

sSet */u() +sSet 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

:sSet */×sSet */sSet */ \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 *sSet_{fin}^{\ast}:

XYcofib(((u(X),*)(*,u(Y)))u(X)×u(Y)). 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 */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} the structure of a closed monoidal category and we write

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

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

Remark

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

[X,Y] n=Hom sSet */(XΔ[n] +,Y) [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

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

is given by

n :(XΔ[n] +fY,YΔ[n] +gZ) (XΔ[n] +X(diag Δ[n]) +X(Δ[n]×Δ[n]) +XΔ[n] +Δ[n] +fidYΔ[n] +gZ). \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 */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} as sSet */sSet^{\ast/}-enriched categories.

Remark

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

:[X 1,Y 1][X 2,Y 2][X 1X 2,Y 1Y 2] \wedge \;\colon\; [X_1,Y_1] \wedge [X_2, Y_2] \longrightarrow [X_1 \wedge X_2, Y_1\wedge Y_2]

by

n :(X 1Δ[n] +f 1Y 1,X 2Δ[n] +f 2Y 2) X 1X 2Δ[n] +iddiag +X 1X 2(Δ[n]×Δ[n]) +X 1X 2Δ[n] +Δ[n] +(X 1Δ[n] +)(X 2Δ[n] +)(f 1) n(f 2) nY 1Y 2 \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 */sSet^{\ast/}-enriched functor category

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

Write

ι:StdSpheressSet fin */ \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}

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

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

  • hom-objects are

    [S std n,S std n+k] StdSpheres{* for k<0 im(S std k[S std n,S std n+k] sSet fin */) otherwise [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 */sSet^{\ast/}-enriched functor

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

(from the category of sSet */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[StdSpheres,sSet */]X \in [StdSpheres,sSet^{\ast/}] to the sequential prespectrum X seqX^{seq} with components

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

and with structure maps

S std 1X n seqσ nX n seqS std 1[X n seq,X n+1 seq] \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 std 1id˜[S std n,S std n+1]X S std n,S std n+1[X n seq,X n+1 seq]. 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 */sSet^{\ast/} enriched equivalence of categories.

(Lydakis 98, prop. 4.3)

The model structures

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

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

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

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

Definition

Write

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

for the functor given on XX by

TX:KΩX(ΣK). T X \colon K \mapsto \Omega X(\Sigma K) \,.

Write

τ:idT \tau \;\colon\; id \longrightarrow T

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

X(K * * ΣK)=X(K) * ΣX(K) τ X(K) * X(ΣK). 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 :[sSet fin */,sSet */][sSet fin */,sSet */] T^\infty \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}] \longrightarrow [sSet^{\ast/}_{fin}, sSet^{\ast/}]

for the functor given by XX by the sequential colimit

T Xlim(Xτ XTXT(τ X)T(TX)). 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:sSet *sSet *Fib \colon sSet^{\ast} \to sSet^{\ast} for any Kan fibrant replacement functor.

Say that the stabilization (spectrification) of XX is

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

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

Definition

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

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

  • a stable cofibration if it has the left lifting property against those morphisms whose value on every KsSet */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 fin */,sSet */] Ly. [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 fin */,sSet */] Ly[sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly} from prop. .

Proposition

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

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

(from the category of sSet */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[StdSpheres,sSet */]X \in [StdSpheres,sSet^{\ast/}] to the sequential prespectrum X seqX^{seq} with components

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

and with structure maps

S std 1X n seqσ nX n seqS std 1[X n seq,X n+1 seq] \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 std 1id˜[S std n,S std n+1]X S std n,S std n+1[X n seq,X n+1 seq]. 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 */sSet^{\ast/} enriched equivalence of categories.

(Lydakis 98, prop. 4.3)

Proposition

The adjunction

(ι *ι *):[sSet fin */,sSet */] Lyι *ι *[StdSpheres,sSet */]() seqSeqPreSpec(sSet) BF (\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 ι *\iota^\ast along the defining inclusion ι\iota of def. and by left Kan extension ι *\iota_\ast along ι\iota, and combined with the equivalence () seq(-)^{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 */,)(sSet^{\ast/}, \wedge) (def. ) is a symmetric monoidal category, [sSet fin *,sSet */][sSet^{\ast}_{fin}, sSet^{\ast/}] canonically becomes symmetric monoidal itself via the induced Day convolution product. We write

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

for this symmetric monoidal category.

Proposition

The smash product on [sSet fin */,sSet */][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 fin */,sSet */](XY,Z)[sSet fin */×sSet fin */,sSet */](X˜Y,Z) [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˜Y(X,Y)X \tilde{\wedge} Y \coloneqq \wedge \circ (X,Y) . Now, (Lydakis 98, def. 5.1) sets

XY *(X˜Y) 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 fin *,sSet */][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

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

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

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

Proposition

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

([sSet fin */,sSet */] Ly, Day). \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 ([sSet fin */,sSet */] Ly, Day) \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 ([sSet fin */,sSet */] Ly, Day)\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right) of prop. are equivalently (symmetric) lax monoidal functors of the form

sSet fin */sSet */ 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 fin */sSet^{\ast/}_{fin}, def. , this is a special casse of a general property of Day convolution, see this proposition.

model structure on functors

model structure on spectra

with symmetric monoidal smash product of 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:

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

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

  • William Dwyer, Localizations, In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 3–28. Kluwer Acad. Publ., Dordrecht, 2004

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

Discussion of the restriction from excisive functors to symmetric spectra includes

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

Further generalization of the model structure for excisive functor, in particular to enriched functors and to a model structure for n-excisive functors for n1n \geq 1 is discussed in

Last revised on March 22, 2016 at 14:42:22. See the history of this page for a list of all contributions to it.