Model structures
2. STABLE MODEL STRUCTURES
Theorem 1.13. The category of symmetric spectra of topological spaces admits the following two level
model structures in which the weak equivalences are those morphismsf :X −→Y such that for all n≥0
the map fn :Xn −→Yn is a weak equivalence of spaces.
(i) In the projective level model structure the cofibrations are the projective cofibrations and a mor- phism f : X −→ Y is a projective level fibration if and only if for every n ≥ 0 the map fn:Yn−→Xn is a Serre fibration.
(ii) In the flat level model structure the cofibrations are the flat cofibrations, and a morphism f :
X −→Y is a flat level fibration if and only if for every n≥0 the map fn : Yn −→ Xn satisfies
the following two equivalent conditions
– the map fn has the right lifting property for all cofibrations of pointed Σn-spaces which are
weak equivalences on underlying spaces;
– map fn is aΣn-fibration and the commutative square
Xn // fn map(EΣn, Xn) map(EΣn,fn) Yn //map(EΣn, Yn)
isΣn-homotopy cartesian. Heremap(EΣn, X)is the space of all maps from the contractible
freeΣn-space to X, with Σn-action by conjugation.
A morphism f : X −→Y is an acyclic fibration in the flat model structure if and only if for all n≥0 the map fn:Xn−→Yn is aΣn-equivariant acyclic fibration.
Moreover, both level model structures are proper, topological and finitely generated, and monoidal with respect to the smash product of symmetric spectra.
[Is there are an injective level model structurefor symmetric spectra of spaces ?] [positive model structures]
2. Stable model structures
Recall from Definition II.4.1 that a morphism f :A−→B if symmetric spectra of simplicial sets is a stable equivalence if for every injective Ω-spectrumX the induced map
[f, X] : [B, X] −→ [A, X] on homotopy classes of spectrum morphisms is a bijection.
For every morphismf :X −→Y the natural morphismλ∗X :X −→Ω(shX) adjoint toλX :S1∧X −→ shX gives rise to a commutative square of symmetric spectra
(2.1) X λ∗X / / f Ω(shX) Ω(shf) Y λ∗Y / /Ω(shY)
Theorem 2.2. The category of symmetric spectra of simplicial sets admits the following threestable model structures in which the weak equivalences are the stable equivalences.
(i) In the projective stable model structure the cofibrations are the projective cofibrations and the fibrations are those projective level fibrations f :X −→Y for which the commutative square (2.1)
is levelwise homotopy cartesian.
(ii) In the flat stable model structurethe cofibrations are the flat cofibrations, and the fibrations are those flat level fibrationsf :X −→Y for which the commutative square (2.1)is levelwise homotopy cartesian.
(iii) In the injective stable model structure the cofibrations are the level cofibrations (i.e., monomor- phisms) and the fibrations are those those injective fibrations f :X −→Y for which the commu- tative square (2.1)is levelwise homotopy cartesian.
Moreover we have:
• All three stable model structures are proper, simplicial and cofibrantly generated.
• The flat and projective stable model structures are even finitely generated and monoidal with respect to the smash product of symmetric spectra.
• In all three cases a morphism is a stable acyclic fibration if and only if it is a level acyclic fibration.
Proof. We reduce the proof of the stable model structures to the level model structures by applying
a general localization theorem of Bousfield, see Theorem 1.8 of Appendix A. In Proposition II.4.21 we constructed a functorQ:SpΣ−→ SpΣwith values in Ω-spectra and a natural stable equivalenceα:A−→
QA. We note that a morphism f : A −→ B of symmetric spectra is a stable equivalence if and only if
Qf : QA −→ QB is a level equivalence. Indeed, since αA : A −→ QA and αB : B −→ QB are stable equivalences,f is a stable equivalence if and only ifQf is. ButQf is a morphism between Ω-spectra, so it is a stable equivalence if and only if it is a level equivalence.
We now apply Bousfield’s Theorem A.1.8 to the injective, flat and projective level model structures. All three level model structures are proper by Theorem 1.9. Axiom (A1) holds since we have a commutative square (2.3) X α // f QX Qf Y α //QY
If f is a level equivalences, thenQf is a stable equivalence between Ω-spectra, hence a level equivalence. Axiom (A2) holds: αQX is a stable equivalence between Ω-spectra, hence a level equivalence. ThenQαX :
QX −→ QQX is a level equivalence since Q takes all stable equivalences, in particular αX, to level equivalences.
We prove (A3) in the projective level model structure. Since the projective fibrations include the flat and injective fibrations, it then also holds in the flat and injective level model structures. So we are given a pullback square V i // f X g W j //Y
of symmetric spectra in which X and Y are Ω-spectra (possibly not levelwise Kan), f is levelwise a Kan fibration andjis a stable equivalence. We showed in part (iv) of Proposition II.4.5 that theniis also a stable equivalence. This proves (A3), and thus Bousfield’s theorem provides three model structures with stable equivalences as weak equivalence and with cofibrations the projective, flat or level cofibrations respectively. Bousfield’s theorem characterizes the fibrations as those level fibrations f : X −→ Y for which the commutative square (2.3) is homotopy cartesian. So it remains to shows that for a morphism f : X −→ Y which is levelwise a Kan fibration the square (2.1) is levelwise homotopy cartesian if and only if the
square (2.3) is levelwise homotopy cartesian.
Corollary 2.4. The following categories are equivalent
• the stable homotopy category, i.e., the homotopy category of injective Ω-spectra of simplicial sets; • the homotopy category of those flat Ω-spectra of simplicial sets for which all Xn are Σn-fibrant
and the maps Xn−→map(EΣn, Xn)are Σn-equivalences;