Question

Hello,

This probably just technical, but anyway:

In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category of chain complexes in non-negative degress. Equivalences are quasi-isom., fibrations are those maps which are surjective in degrees $n \ge 1$ (notice the $1$!).

Then they note: "After the fact, it turns out that the cofibrations are those monomorphisms of chain complexes having degreewise projective cokernels".

It seems this is not correct as stated. Maybe if it would say $n \ge 0$ upstairs it would be correct, but we do have $n \ge 1$ (corresponding to the picture of simplicial abelian groups).

Is it a typo or I don't understand something? Is there a nice description of cofibrations in this case?

Thank you, Sasha

Update: I was just confused with homological/cohmological indexing conventions. One needs either chains in positive degrees, or cochains in negative degrees. I by mistake used positive degrees as they, but with cochains - by habit.

Answer 1

The definition is correct. (That is, it correctly describes a model structure, and this is a commonly used model structure.)

Note that for nonnegatively graded chain complexes there is no chance of requiring all fibrations to be surjective in degree zero because (given the requirement that every map can be factored as fibration composed with weak equivalence) this would force all maps to induce surjections in $H_0$.

For chain complexes graded by $\mathbb Z$ there is indeed a model structure with surjections as the fibrations. Again, the cofibrations work out to be the monomorphisms with degreewise projective cokernel.

All of this generalizes to chain complexes of $R$-modules, except that in the $\mathbb Z$-graded case it turns out that for general $R$ not every monomorphism with projective cokernel is a cofibration.

Your mention of simplicial abelian groups reminds me to point out that Dold-Kan gives an equivalence of categories, so that a model structure on simplicial groups corresponds precisely to such a structure on nonnegatively graded chain complexes. This also generalizes to $R$-modules rather than abelian groups. The relevant model structure on simplicial modules is such that a map is a fibration [resp. weak equivalence] if and only if the forgetful functor to simplicial sets takes it to a fibration [resp. weak equivalence].

Answer 2

This is the same way that the fibrations are described in this paper by Dwyer and Spalinski. Section 7 contains the relevant material. Here the cofibrations are monomorphisms in degrees $n \geq 0$ with projective cokernal.