TOOLS FROM MODEL CATEGORY THEORY

Proof. Model category axiom MC1 (limits and colimits) holds by hypothesis. Model category axioms

MC2 (saturation) and MC3 (closure properties under retracts) are clear. One half of MC4 (lifting properties) holds by the definition of cofibrations in D.

The proof of the remaining axioms uses the transfinite small object argument (Lemma 1.1), which applies because of the hypothesis about the smallness of the domains. We begin with the factorization axiom, MC5. Every map inLI andLJ is a cofibration in Dby adjointness. Hence everyLI-cofibration or

LJ-cofibration is a cofibration inD. By adjointness and the fact that I is a generating set of cofibrations for C, a map is LI-injective precisely when the map becomes an acyclic fibration inC after application of

R, i.e., an acyclic fibration inD. Hence the small object argument applied to the setLI gives a (functorial) factorization of any map inDas a cofibration followed by an acyclic fibration.

The other half of the factorization axiom, MC5, needs the hypothesis. Applying the small object argument to the set of maps LJ gives a functorial factorization of a map in D as an LJ-cell complex followed by aLJ-injective. SinceJ is a generating set for the acyclic cofibrations inC, theLJ-injectives are precisely the fibrations among the D-morphisms, once more by adjointness. We assume that everyLJ-cell complex is a weak equivalence. We noted above that every LJ-cofibration is a cofibration inD. So we see that the factorization above is an acyclic cofibration followed by a fibration.

It remains to prove the other half of MC4, i.e., that any acyclic cofibrationA−→B in Dhas the left lifting property with respect to fibrations. In other words, we need to show that the acyclic cofibrations are contained in theLJ-cofibrations. The small object argument provides a factorization

A−→W −→B

with A −→ W a LJ-cofibration andW −→ B a fibration. In addition, W −→ B is a weak equivalence sinceA−→B is. SinceA−→B is a cofibration, a lifting in

A // W ∼ B > > B

exists. ThusA−→B is a retract of aLJ-cofibration, hence it is aLJ-cofibration. In cofibrantly generated model categories fibrations can be detected by checking the right lifting property against a set of maps, the generating acyclic cofibrations, and similarly for acyclic fibrations. This is in contrast to general model categories where the lifting property has to be checked against the whole class of acyclic cofibrations. Similarly, in cofibrantly generated model categories, the pushout product axiom and the monoid axiom only have to be checked for a set of generating (acyclic) cofibrations:

Lemma1.5. LetC be a cofibrantly generated model category endowed with a closed symmetric monoidal

structure. If the pushout product axiom holds for a set of generating cofibrations and a set of generating acyclic cofibrations, then it holds in general.

Proof. For the first statement consider a map i:A −→ B in C. Denote by G(i) the class of maps

j:K−→Lsuch that the pushout product

A∧L∪A∧KB∧K −→ B∧L

is a cofibration. This pushout product has the left lifting property with respect to a map f:X −→Y if and only ifj has the left lifting property with respect to the map

p: [B, X] −→ [B, Y]×[A,Y][A, X].

Hence, a map is in G(i) if and only if it has the left lifting property with respect to the map p for all

f:X −→Y which are acyclic fibrations inC.

G(i) is thus closed under cobase change, transfinite composition and retracts. If i : A −→ B is a generating cofibration, G(i) contains all generating cofibrations by assumption; because of the closure properties it thus contains all cofibrations, see Lemma 1.1. Reversing the roles of i and an arbitrary

cofibrationj :K −→L we thus know thatG(j) contains all generating cofibrations. Again by the closure properties, G(j) contains all cofibrations, which proves the pushout product axiom for two cofibrations. The proof of the pushout product being an acyclic cofibration when one of the constituents is, follows in

the same manner.

We now spell out the small object argument for symmetric spectra.

Theorem 1.6 (Small object argument). Let I be a set of morphisms of symmetric spectra based on

simplicial sets. Then there exists a functorial factorization of morphisms as I-cell complexes followed by I-injective morphisms.

Proof. In the first step we construct a functorFfrom the category of morphisms of symmetric spectra

to symmetric spectra as follows. Given a morphism f :X −→Y and a morphismi:Si−→Ti in the setI we letDidenote the set of all pairs (a:Si−→X, b:Ti−→Y) of morphisms satisfyingf a=bi, i.e., which make the square

Si a _// i X f Ti b //Y commute. We define F(f) as the pushout in the diagram

W i∈I W DiSi ∨a _// ∨i X j W i∈I W DiTi //F(f)

The morphisms b : Ti −→ Y and f : X −→ Y glue to a morphism p: F(f) −→ Y such that pj = f. The factorization we are looking for is now obtained by iterating this construction infinitely often, possibly transfinitely many times.

We define functorsFn:Ar(SpΣ)−→SpecΣand natural transformationsX −→jn Fn(f)−→pn Y for every ordinalnby transfinite induction. We start withF0(f) =X,j0= Id andp0=f. For successor ordinal we setFn+1(f) =F(pn:Fn(f)−→Y) with the morphismsjn+1=j◦jnrespectivelypn+1=p(pn). For limit ordinalsλwe setFλ_{(f) = colim}

µ<λFµ(f) with morphisms induced by thejµ andpµ. By construction, all morphismsjn :X −→Fn(f) areI-cell complexes.

We claim that there exists a limit ordinal κ, depending on the setI, such that for every morphismf

the mappκ:Fκ(f)−→Y isI-injective. Thenf =pκjκis the required factorization.

We prove the claim under the simplifying hypothesis that for each morphism i ∈ I the sourceSi is

finitely presented as a symmetric spectrum, i.e., for every sequenceZ0 −→Z1−→Z2−→. . . the natural map

colimn SpΣ(Si, Zn) −→ SpΣ(Si,colimnZn)

is bijective. In that case, the first infinite ordinal ω will do the job. Indeed, Fω_{(f) is the colimit over the} sequence

X=F0(f) j1

−→F1(f) j2

−→F2(f)· · · .

Given a morphismi∈I and a lifting problem

(1.7) Si a // i Fω(f) pω Ti _b //Y

1. TOOLS FROM MODEL CATEGORY THEORY 143