ETAPA DE AISLAMIENTO DE Azotobacter chroococcum nativo

just defined has the structure of a triangulated category. We first develop some tools needed in these later sections.

Proposition 1.6. Let X be an injective spectrum.

(i) For every injective morphism i : A −→ B of symmetric spectra the map map(i, X) : map(B, X)−→map(A, X) is a Kan fibration of simplicial sets. If in additioni is a level equiva- lence, then map(i, X)is a weak equivalence.

(ii) For every symmetric spectrumBthe function spacemap(B, X)is a Kan complex and the homotopy relation for morphisms from B toX is an equivalence relation.

(iii) For every n≥0 the simplicial setXn is a Kan complex.

Proof. (i) We have to check that map(i, X) has the right lifting property with respect to every injective

weak equivalence j:K−→Lof pointed simplicial sets. By the adjunction between the smash pairing and mapping spaces, a lifting problem in the form of a commutative square

K // j map(B, X) map(i,X) L //map(A, X)

corresponds to a morphismK∧B∪K∧AL∧A−→X, and a lifting corresponds to a morphismL∧B−→X which restricts to the previous morphism along the ‘pushout product’ mapj∧i:K∧B∪K∧AL∧A−→L∧B. Sincejis an injective weak equivalence andiis injective, the pushout product morphismj∧iis an injective level equivalence of symmetric spectra. So the lifting exists since we assumed thatX is injective.

The second part is very similar. Ifiis injective and a level equivalence, then for every injective morphism

j :K−→L(not necessarily a weak equivalence) of pointed simplicial sets, the pushout product mapj∧iis an injective level equivalence of symmetric spectra. So map(i, X) has the right lifting property with respect to all injective morphisms of pointed simplicial sets.

Part (ii) is the special case of (i) where A is the trivial spectrum so that map(A, X) is a one-point simplicial set. Vertices of the simplicial set map(B, X) correspond bijectively to morphismsB −→ X in such a way that 1-simplices correspond to homotopies. So the second claim of (ii) follows since in every Kan complex, the relationx∼y on vertices defined by existence of a 1-simplexzwithd0z=xandd1z=y is an equivalence relation.

The simplicial setXn is naturally isomorphic to the mapping space map(FnS0, X) with source the free symmetric spectrum generated byS0 _{in level n. So (iii) is a special case of (ii).}

We now get a criterion for level equivalence by testing against injective spectra.

Proposition1.7. A morphismf :A−→Bof symmetric spectra of simplicial sets is a level equivalence

if and only if for every injective spectrum X the induced map[f, X] : [B, X]−→[A, X]on homotopy classes of morphisms is bijective.

Proof. Suppose first thatf is a level equivalence. We replacef by the inclusion ofAinto the mapping

cylinder off, which is homotopy equivalent toB. This way we can assume without loss of generality thatf

is injective. By part (i) of Proposition 1.6 the map map(f, X) : map(B, X)−→map(A, X) is then a weak equivalence of simplicial sets, so in particular a bijection of components. Sinceπ0map(B, X)∼= [B, X], and similarly forA, this proves the claim.

Now suppose conversely that [f, X] : [B, X]−→[A, X] is bijective for every injective spectrumX. IfK

is a pointed Kan complex andm≥0, then the co-free symmetric spectrumRmKof Example 1.4 is injective. The adjunction for morphisms and homotopies provides a natural bijection [A, RmK]∼= [Am, K]sset* to the

based homotopy classes of morphisms of simplicial sets. So for every Kan complex K, the induced map [fm, X] : [Bm, K]−→[Am, K] is bijective, which is equivalent tofn being a weak equivalence of simplicial sets. Since this holds for allm, the morphismf is a level equivalence.

The next lemma can be used to recognize certain morphisms as homotopy equivalences, and thus as isomorphism in the stable homotopy category.

Proposition 1.8. (i) Every level equivalence between injective spectra is a homotopy equivalence.

(ii) Every π∗-isomorphism betweenΩ-spectra is a level equivalence.

Proof. (i) Let f : X −→ Y be a level equivalence between injective spectra. Using the mapping

cylinder construction, f can be factored as a monomorphism followed by a homotopy equivalence. So we can replaceY by the mapping cylinder and assume without loss of generality thatf is also a monomorphism. By Proposition 1.6 (i) the induced map map(f, X) : map(Y, X)−→map(X, X) is a weak equivalence and Kan fibration, thus surjective on vertices. So there is a morphismg:Y −→X satisfyinggf = IdX.

Also by Proposition 1.6 (i) the induced map map(f, Y) : map(Y, Y)−→ map(X, Y) is a weak equiv- alence and Kan fibration. Since moreover map(f, Y) takes the vertices f g and IdY of map(Y, Y) to the same vertex (namely f ∈map(X, Y)), they can be joined by a 1-simplex in map(Y, Y), i.e., a homotopy of spectrum morphisms.

(ii) For every Ω-spectrumX and allk, n≥0, the canonical mapπkXn−→πk−nX is a bijection. So if

f :X −→Y is aπ∗-isomorphism between Ω-spectra, then for every n≥0, the morphism fn :Xn −→Yn induces a bijection of path components and isomorphisms of homotopy groups in positive dimensions, based at the distinguished basepoint of Xn. In particular, fn restricts to a weak equivalence between the components containing the distinguished basepoints. SinceXn and Yn are loop spaces, their various path components are all weakly equivalent, and so fn restricts to a weak equivalence on every path component ofX, i.e.,fn is a weak equivalence of simplicial sets for everyn≥0.

Theorem 1.9. For every π∗-isomorphismf :A−→B between symmetric spectra and every injective

Ω-spectrumX the induced map on homotopy classes [f, X] : [B, X]−→[A, X]is a bijection.

Proof. We use the functor R∞ introduced in (4.43) of Chapter I. Since X is an Ω-spectrum, the morphism λ∗ : X −→ RX = Ω(shX) is a level equivalence, and so are all other morphisms in the se- quence (4.43) whose colimit isR∞_{X. Thus also the morphismλ}∞

X :X−→R∞X is a level equivalence, and

R∞X is again an Ω-spectrum. SinceX is an injective spectrum the map [λ∞_X, X] : [R∞X, X]−→[X, X] is bijective by Proposition 1.7. So there exists a morphismr:R∞X −→X such that the composite rλ∞ is homotopic to the identity of X (the other composite need not be homotopic to the identity ofR∞X).

The functorR∞ preserves the homotopy relation, so we can define a natural transformation [A, X] −→ [R∞A, X], [ϕ]7→[r◦R∞ϕ].

There also is a natural transformation [R∞A, X]−→[A, X] in the other direction given by precomposition with λ∞_A :A−→R∞A. Since ris a retraction (up to homotopy) to λ∞_X, the composite of the two natural maps is the identity on [A, X]. In other words, for fixed injective Ω-spectrum X, the functor [−, X] is a retract of the functor [R∞(−), X].

Now suppose thatf :A−→B is aπ∗-isomorphism. We assume first that bothA andB are levelwise

Kan complexes. In [prove] we established a natural isomorphism πk(R∞A)m ∼=πk−mA (but beware that

R∞A is not an Ω-spectrum unless A is semistable). So R∞f : R∞A −→ R∞B is a level equivalence [homotopy groups at other basepoints ?]. So by Proposition 1.7 the map [R∞f, X] : [R∞B, X]−→[R∞A, X] is bijective. Since this has [f, X] : [B, X]−→[A, X] as a retract, the latter is bijective.

In general we apply the functors ‘geometric realization’ and ‘singular complex’ to the morphism f :

A −→ B to replace it by a level equivalent morphism whose source and target are levelwise Kan. Since [−, X] takes level equivalences to bijections, the general case follows from the special case. If X is an Ω-spectrum, then so is the shifted spectrum shX. The left adjoint S0.0 to shifting (see Example I.4.37) preserves level equivalences and level injections, so shifting also preserves the property of being injective. Moreover, shifting preserves homotopies since sh(∆[1]+∧X) = ∆[1]+∧ shX. IfX is an Ω-spectrum and levelwise Kan, then so is the loop spectrum ΩX, and the functor Ω preserves injective spectra and the homotopy relation. Moreover, for every Ω-spectrum X the natural map

2. ADDITIVE STRUCTURE 89