model structure for excisive functors in nLab

Context

Model category theory

Stable Homotopy theory

• Idea
• Definition
• The underlying categories
• The model structures
• Properties
• Relation to BF-model structure on sequential spectra
• Symmetric monoidal smash product
• Related concepts
• References

Definition

Write

• sSet for the category of simplicial sets;

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

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

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}:

X∧Y≔cofib(((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.

Definition

Write S std 1≔Δ[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:StdSpheres⟶sSet 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.

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

and with structure maps

S std 1∧X n seq⟶σ 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 1⟶id˜[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.

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:K↦ΩX(ΣK). T X \colon K \mapsto \Omega X(\Sigma K) \,.

Write

τ:id⟶T \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 (Σ⊣Ω)(\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 ∞X≔lim⟶(X⟶τ XTX⟶T(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:X→Yf \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 K∈sSet */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 K∈sSet */K \in sSet^{\ast/} is a Kan fibration.

Proposition

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

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

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

and with structure maps

S std 1∧X n seq⟶σ 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 1⟶id˜[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.

Proposition

The adjunction

(ι *⊣ι *):[sSet fin */,sSet */] Ly⟶ι *⟵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. .

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.

Proof

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

[sSet fin */,sSet */](X∧Y,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

X∧Y≔∧ *(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.

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}).

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) \,.

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).