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I noticed that cell complex was missing, so I created it
The current definition is “a transfinite composition of pushouts of the generating cofibrations”. Should this be “a transfinite composition of pushouts of coproducts of the generating cofibrations”? I think equivalence of these definitions requires AC.
That’s the definition as stated in Hovey 1999 Def. 2.1.9 and Hirschhorn 2002 Def. 10.5.8.
The use of non-constructive methods is typically tacitly understood here (already the existence of the classical model structure on topological spaces needs the axiom of choice).
But I have now expanded a little in the entry and added a Remark on this point (here).
Re #3: The current version says:
(Note that the axiom of choice is needed already for basic statements in this context, like the very existence of the classical model structure on topological spaces, see there this Lemma).
Is “needed” actually correct? Do we have a proof that the existence of the Serre–Quillen model structure implies the axiom of choice?
Both functorial factorizations in the Serre–Quillen model structure, for example, can be constructed using explicit formulas. The existence of liftings might require the axiom of choice, but do we have a proof of this?
No, I just mean that the Lemma I pointed to is used in the existence proof.
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