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Theorem 4.12. The functorγ:SpΣ−→ SHC is a localization of the category of symmetric spectra at

the class of stable equivalences. In particular, for every functorε:SpΣ_{−→ Cwhich takes stable equivalences}

to isomorphisms, then there exists a functor ε:SHC −→ C, unique up to preferred natural isomorphism, such that ε◦γ is naturally isomorphic to ε.

Proof. We start by showing that γ takes stable equivalences to isomorphisms. By definition ofγf

we have the homotopy commutative diagram (4.9). So if f :A−→B is a stable equivalence, thenγf is a stable equivalence between injective Ω-spectra and therefore a homotopy equivalence. In other words, γf

is an isomorphism inSHC.

Next make some observations about functorsε:SpΣ−→ E which invert stable equivalences. For every spectrum A, the projectionπ: ∆[1]+_{∧A−→A is a homotopy equivalence, hence a stable equivalence, so}

ε(π) is an isomorphism. The two end inclusionsi0, i1:A−→∆[1]+∧Asatisfyπ◦i0= IdA=π◦i1, so we have

ε(π)◦ε(i0) = Idε(A) = ε(π)◦ε(i1). Sinceε(π) is an isomorphism, we deduceε(i0) =ε(i1).

Suppose now thatf, g:A−→B are homotopic morphisms via some homotopy H : ∆[1]+_∧A−→B. Then

ε(f) = ε(H)◦ε(i0) = ε(H)◦ε(i1) = ε(g).

In other words, every functor ε : SpΣ _{−→ E which takes stable equivalences to isomorphisms also takes} homotopic maps to the same morphisms.

Now we show that for every functor ε : SpΣ _{−→ E which takes stable equivalences to isomorphisms} there is a functor ε:SHC −→ E such that εγ is naturally isomorphic to ε. This proves that the functor

− ◦γ: Hom(SHC,E)−→Homst. equi.(SpΣ_{,E) is dense (essentially surjective on objects). We simply define}

ε:SHC −→ E on objects byε(A) =ε(A) and on morphisms via representatives by ε[f :A−→B] =ε(f). This will automatically be a functor. If we apply the functor εto the stable equivalencepA:A−→γAwe get a natural isomorphism inE

ε(pA) : ε(A) −→ ε(γA) = (εγ)(A).

It remains to show that precomposition with γ is fully faithful. So we consider two functors F, G :

SHC −→ E and have to show that

− ◦γ : Nat(F, G) −→ Nat(F γ, Gγ)

is bijective. We define the inverse map K : Nat(F γ, Gγ) −→ Nat(F, G) as follows. Given a natural transformationτ :F γ−→Gγ of functorsSpΣ_{−→ E we define the natural transformationK(τ) :F −→G} of functors SHC −→ E as the restriction of τ to injective Ω-spectra. This makes sense because we had insisted earlier that γX=X andpX= Id wheneverX is an injective Ω-spectrum.

We haveK(τ)◦γ=τ as natural transformations becauseγ(γX) =γX andpγX= Id (again becauseγ is the identity on injective Ω-spectra). We also haveK(ϕ◦γ) =ϕfor a natural transformationϕ:F −→G, again because γX =X for every injective Ω-spectrum. So K is indeed inverse to precomposition withγ,

which finishes the proof.

The next proposition makes precise in which way the suspension of a symmetric spectrum ‘is’ the shift in the stable homotopy category and how homotopy cofibre and homotopy fibre sequences give rise to distinguished triangles inSHC.

For any symmetric spectrumA we have a diagram of morphisms of symmetric spectra

S1_∧A pS1∧A // S1_∧p A γ(S1_∧A) ΦA S1_∧γA λγA / /sh(γA)

in which all solid arrows are stable equivalences. Since sh(γA) is an injective Ω-spectrum, there exists unique homotopy class of morphisms ΦA : γ(S1∧A)−→sh(γA) which makes the entire square commute up to homotopy. The morphism ΦA is a stable equivalence between injective Ω-spectra, thus an isomorphism

(4.13) ΦA : γ(S1∧A) ∼= (γA)[1]

in the stable homotopy category.

Dually, the functorγcommutes with taking functions out of a finite simplicial set, so in particular with loops. Indeed, if Kis a finite pointed simplicial set andA a symmetric spectrum which is levelwise a Kan complex, then the morphism (pA)K :AK−→(γA)K is a stable equivalence by part (v) of Proposition 4.5. Since (γA)K _{is an injective Ω-spectrum, there is a unique homotopy class of morphisms Ψ}

A,K:γ(AK)−→ (γA)K_{such that Ψ}

A,K◦pAKis homotopic to (pA)K. This morphism is a stable equivalence between injective Ω-spectra, thus a homotopy equivalence. An important special case is K =S1_{, which yields a preferred} homotopy class of homotopy equivalence ΨA,S1 :γ(ΩA)−→Ω(γA).

The isomorphisms ΦA :γ(S1_{∧A)−→ (γA)[1] and ΨA,K : γ(A}K_{)−→ (γA)}K _{are natural in SHC as} functors ofA, which is a consequence of the uniqueness properties.

Proposition 4.14. (i) Let f : A −→ B be a morphism of symmetric spectra. Then the functor

γ:SpΣ_{−→ SHC takes the sequence}

A−→f B−→i C(f)−→p S1∧A

to a distinguished triangle in the stable homotopy category after identifying γ(S1∧A) with(γA)[1]via φA. (ii) Let f : A −→ B be a morphism of symmetric spectra which are levelwise Kan complexes. Then the functor γ:SpΣ_{−→ SHC takes the sequence}

ΩB−→i F(f)−→p A−→f B

to a distinguished triangle in the stable homotopy category after making the identifications

(γ(ΩB))[1] −−−−−→ΨB,S1[1] Ω(γB)[1] λ

∗

γB

←−−− γB .

(iii) The functor γ commutes with arbitrary coproducts and finite products. Thus in particular the stable homotopy category has arbitrary coproducts.

Proof. (iii) Given a family {Ai}i∈I of symmetric spectra, then the wedge of the stable equivalences

pi:Ai−→γ(Ai) is a stable equivalence by Proposition 4.5 (i). Sinceγ(WiAi) is an injective Ω-spectrum, there is a unique homotopy class of morphism g : W

iγ(Ai) −→ γ(

W

iAi) whose restriction with

W

ipi is homotopic to the stable equivalence pW_A

i. The morphismg is then also a stable equivalence, and so for

every injective Ω-spectrumX the induced map on homotopy classes [g, X] : [γ(_

i

Ai), X] −→[_ i

γ(Ai), X]

is bijective. The target is isomorphic to the productQ

i[γ(Ai), X], which shows that the injective Ω-spectrum

γ(W

iAi) has the universal property of a coproduct of the objectsγ(Ai).

The proof thatγ preserves finite products is similar but slightly easier because products in the stable homotopy category are given by pointset level products. We consider the case of two factors. By the same reasoning as before there is a unique homotopy class of morphism h: γA×γB −→γ(A×B) satisfying

h(pA×pB) =pA×B, andhis a stable equivalence. But now both sides are injective Ω-spectra, so the stable equivalence is even a homotopy equivalence, i.e., an isomorphism inSHC. 4.2. The homotopy groups ofγA. We now have a way of associating to every symmetric spectrum an object of the stable homotopy category, via the functorγ:SpΣ_{−→ SHC. However, the functor depends} on an abstract construction which produces a stable equivalence to an injective Ω-spectrum. This does not make it transparent what ‘happens’ to a symmetric spectrum during this passage, and it not clear how basic invariants like stable homotopy groups change in this process.

4. STABLE EQUIVALENCES 107