Symmetric Monoidal Categories for Operads - Index of

11.1 Recollections: The Language of Model Categories 161
If we assume further that f is an acyclic cofibration, then the fibration
q is also acyclic by the two-out-of-three axiom. By axiom M4.i, which is
tautologically satisfied in A, we have a morphism s so that sf = j and
ps = id, from which we deduce that f is a retract of j. Since retracts inherit
left lifting properties, we obtain that f has the left lifting property with
respect to fibrations as well. This argument proves that the lifting axiom
M4.ii is also satisfied in A and achieves the proof of the theorem. ⊓⊔
In our constructions, we use the following proposition to check the conditions
of theorem 11.1.13:
11.1.14 Proposition. The conditions of theorem 11.1.13 are satisfied under
the sufficient assumptions that:
(1) The functor U : A→X preserves colimits over non-empty ordinals;
(2) For any pushout
F (C) A ,
F (i)
F (D) B
the morphism U(f) forms a cofibration, respectively an acyclic cofibration,
in X if i is so.
In this situation, the functor U : A→X preserves cofibrations in addition to
create weak-equivalences and fibrations.
Proof. Let j : A → B be a relative F I-cell (respectively, F J -cell) complex
in A. The assumptions imply that the morphism U(j) splits into a colimit
U(A)=U(B0)→ ...→U(Bλ−1) U(jλ)
−−−→ U(Bλ)→···→colimλ