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added pointer to the original statement in
Started (here) a paragraph on the characterization of the cofibrations of simplicial groups. I gather that
but where is this actually proven?
In Goerss & Jardine (1999) Cor. 1.10 there is (only) proof that
which of course implies
but where is the proof of the converse implication “”?
I understand that Quillen would have said “free” for “almost free”, but even so I haven’t spotten this in either his “Homotopical Algebra” nor “Rational Homotopy Theory”.
On the other hand, I see people state it as a fact, eg. Baues (1999, p. 27) or the nice review of Speirs (2015, p. 54).
Re #4: This is a special instance of the fact that in any cofibrantly generated model category, cofibrations are codomain retracts of cellular maps. Cellular maps in their turn are transfinite compositions of cobase changes of generating cofibrations.
Cellular maps of simplicial groups are precisely almost free maps.
Thanks. This must be citable from somewhere?
Re #6: Yes, this is part of the statement of the small object argument.
For example, see Proposition 10.5.16 in Hirschhorn.
It is also explained in detail Joyal’s CatLab article Weak factorisation systems (joyalscatlab), Theorem 3.17.
No, it’s a statement about the cofibrant generation of simplicial groups, where do you take this from?
Re #8: Transferring a cofibrantly generated model structure along a right adjoint functor produces a cofibrantly generated model structure. This is part of the transfer theorem, see the original version in Crans (Theorem 3.3), or Theorem 11.3.2 in Hirschhorn.
Quillen’s book Homotopical Algebra already defines the model structure on simplicial groups as the transferred model structure, see Theorem II.2.2.
I see, thanks, so I guess the claim is that almost free maps are the composites of pushouts of the .
Once that is true, I understand the statement. (Though I don’t see what codomain retracts have to do with it, it seems we need to be talking about plain retracts.)
Now why is it true that almost free maps are the the relative -cell complexes? That sounds very plausible, as these arise from iteratively amalgamating free group generators — but it’s not immediate (to me) that exactly those compatibility conditions entering the definition of “almost free maps” are satisfied by these cell complexes.
(Though I don’t see what codomain retracts have to do with it, it seems we need to be talking about plain retracts.)
A codomain retract is a special case of a plain retract, in which the map on domains is the identity map. Thus, A→B is a codomain retract of A→C if B is a retract of C, and the retraction C→B makes the corresponding triangle commute.
What the small object argument gives you is not only a retract, but in fact a codomain retract, which is a stronger (and better) property.
Now why is it true that almost free maps are the the relative {F(∂Δ[n])→F(Δ n)} n∈ℕ-cell complexes?
This is precisely the content of Proposition V.1.9 in Goerss–Jardine, which starts with an almost free map and presents it (inductively on the skeleton) as a (countable) transfinite composition of cobase changes of maps of the form {F(∂Δ[n])→F(Δ[n])}.
This is precisely the content of Proposition V.1.9
That’s the other direction: Their proposition states that given an almost free map, then it is such a pushout and hence a cofibration. The question in #4 is for proof of the converse.
It is plausible that what you indicate works, and it should be a matter of carefully going through the combinatorics. On the other hand I wonder why Goerss & Jardine didn’t state the converse if they were so close to proving it, while those authors who I see stating it don’t prove it.
In any case, I have made a note of what we have so far: here.
That’s the other direction: Their proposition states that given an almost free map, then it is such a pushout and hence a cofibration. The question in #4 is for proof of the converse.
For the converse, observe that the class of almost free maps is weakly saturated, since colimits of simplicial groups are computed objectwise.
Thus, to show that cofibrations are contained in almost free maps, it suffices to show that the generating cofibrations F(∂Δ^n)→F(Δ^n) are almost free maps, which is clear.
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