Model structures
1. LEVEL MODEL STRUCTURES 129 (iii) In the injective level model structure the cofibrations are the level cofibrations (i.e., monomor-
phisms) and the injective fibrations are those morphisms which have the right lifting property with respect to all morphisms which are simultaneously level equivalences and monomorphisms. Moreover we have:
• All three level model structures are proper, simplicial and cofibrantly generated.
• The flat and projective level model structures are even finitely generated and monoidal with respect to the smash product of symmetric spectra.
Proof. The category of symmetric spectra of simplicial sets has all set-indexed limits and colimits, the
level equivalences satisfy the 2-out-of-3 property and in all three cases the classes of cofibrations, fibrations and weak equivalences are closed under retracts. So it remains to prove the factorization and lifting axioms. As usual we construct the factorizations using Quillen’s small object argument. We first defined the respective classes Iprojlv , If latlv and Iinjlv of generating cofibrations and Jprojlv , Jf latlv and Jinjlv of generating acyclic cofibrations. As generating projective cofibrations we take
Iprojlv = Fn∂∆[m]+ −→ Fn∆[m]+ n,m≥0
whereFnis the free symmetric spectrum generated by a pointed simplicial set in leveln, see Example I.2.12. SinceFn is left adjoint to evaluation at leveln, a morphismf :X −→Y of symmetric spectra has the right lifting property with respect to Iprojlv if and only if for every n≥0 the mapfn :Yn −→Xn has the RLP for the boundary inclusions ∂∆[n]−→∆[n], i.e., if it is a Kan fibration and a weak equivalence. In other words, precisely the morphisms which are both level equivalences and projective level fibrations enjoy the right lifting property with respect to Ilv
proj. As generating flat cofibrations we take
If latlv = n Gm(Σm/H×∂∆[n]) + −→ Gm(Σm/H×∆[n]) +o n,m≥0, H≤Σn
where Gm is the semifree symmetric spectrum generated by a pointed Σm-simplicial set in level m, see Example I.2.13. Since Gm is left adjoint to evaluation at level m with values in Σm-simplicial sets, a morphismf :X−→Y of symmetric spectra has the right lifting property with respect toIf latlv if and only if for every m ≥0 and every subgroup H of Σn the map (fm)H : (Ym)H −→ (Xm)H on H-fixed points of fm is a Kan-fibration and weak equivalence of simplicial sets. By Proposition 1.9 of Appendix A this is equivalent to the property that fn is simultaneously a Σn-fibration, a weak equivalence on underlying simplicial sets and the square above is Σn-homotopy cartesian; in other words, precisely the flat level fibrations defined in (ii) above enjoy the right lifting property with respect to Ilv
f lat.
Letf :X −→Y be a flat level fibration. IfHis a subgroup of Σnthen the semifree symmetric spectrum
Gn(Σn/H)+ is flat. Since the flat model structure is simplicial, the induced map on mapping spaces (Xn)H∼= map(Gn(Σn/H)+, X)−→map(Gn(Σn/H)+, Y)∼= (Yn)H
is a Kan fibration. Since this holds for all subgroup, the map fn:Xn−→Yn is a Σn-fibration.
Suppose thatX is fibrant in the flat level model structure. LetLbe any (unbased) Σm-simplicial set. Then the projectionEΣn×L−→Lis Σn-equivariant and a weak equivalence of underlying simplicial sets. So the induced map of semifree symmetric spectra
Gn(EΣn×L)+−→GnL+
is a level equivalence between flat symmetric spectra. Since the flat model structure is simplicial, the induced map on mapping spaces
map(GnL+, X)−→map(Gn(EΣn×L)+, X) is a weak equivalence. By adjointness that map is isomorphic to
mapΣn(L, Xn)−→mapΣn(EΣn×L, Xn)∼= mapΣn(L,map(EΣn, X
where map(EΣn, Xn) is the space (i.e., simplicial set) of all morphisms fromEΣn toX with conjugation action by Σn. If we specialize toL= Σn/H for a subgroupH of Σn we see that the map
Xn−→map(EΣn, Xn)
is a weak equivalence on H-fixed points, so it is an equivariant equivalence.
To define the generating injective cofibrations we choose one representative for each isomorphism class of pairs (B, A) consisting of acountable symmetric spectrumB and a symmetric subspectrumA.
We still have to show that the level model structures are simplicial and that the flat and projective level model structures are monoidal with respect to the internal smash product of symmetric spectra. So we have to verify various forms of the pushout product property. We recall that the pushout product of a morphism i :K −→L of pointed simplicial sets or symmetric spectra and a morphism j : A−→ B of symmetric spectra is the morphism
i∧j : L∧A ∪K∧A K∧B −→ L∧B .
The first proposition below is about smash products of simplicial sets with symmetric spectra, and it says that various model structures of symmetric spectra are simplicialmodel structures. The next proposition is about internal smash products of symmetric spectra, and it says that various flat and projective (but not injective) model structures of symmetric spectra are monoidalmodel structures.
Proposition 1.10. Let i : K −→ L be a morphism of pointed simplicial sets and j : A −→ B a morphism of symmetric spectra.
(i) If iis injective andj a level cofibration, flat cofibration respectively projective cofibration, then the pushout product i∧j is also a level cofibration, flat cofibration respectively projective cofibration.
(ii) If i is an injective weak equivalence of simplicial sets, andj is a level cofibration (i.e, monomor- phism), then i∧j is also a level equivalence of symmetric spectra.
(iii) If i is injective and j a level cofibration (i.e, monomorphism) and a level equivalence, π∗-
isomorphism respectively stable equivalence of symmetric spectra, then i∧j is also a level equiva- lence,π∗-isomorphism respectively stable equivalence.
Thus the injective, flat and projective level model structures are simplicialmodel categories.
Proof. For every pointed simplicial set K and symmetric spectrum A the smash product K∧L is
naturally isomorphic to the smash product of the suspension spectrum Σ∞K with A. The suspension spectrum functor takes injective maps of simplicial sets to projective cofibrations (see Proposition 1.5 (iii) for m = 0) and it takes weak equivalences to level equivalences. So this proposition is a special case of
Proposition 1.11 below.
Proposition 1.11. Let i:K−→Landj :A−→B be morphisms of symmetric spectra.
(i) If i is a level cofibration andj is a flat cofibration, theni∧j is a level cofibration.
(ii) If both iandj are flat cofibrations, then so isi∧j.
(iii) If both iandj are projective cofibrations, then so isi∧j.
(iv) If iis a level cofibration,j a flat cofibration and one ofiorj a level equivalence,π∗-isomorphism
respectively stable equivalence, then i∧j is also a level equivalence, π∗-isomorphism respectively
stable equivalence.
Thus the flat and projective level model structures are monoidalmodel categories with respect to the smash product of symmetric spectra.
Proof. Check on generators.
[State all adjoint forms of the simplicial and monoidal axiom]
Definition1.12. A morphismf :K−→Lof Σn-spaces is a Σn-fibration(respectively Σn-equivalence)
if the induced map onH-fixed pointsfH:KH −→LH is a Serre fibration (respectively weak equivalence) for all subgroups H of Σn.