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(First, an apology: I accidentally posted this in the wrong place, under “MathForge general discussions.” I tried to delete it from there but couldn’t figure out how to do so. Could someone help me?)
My question relates to the entry
http://ncatlab.org/nlab/show/derived+functor#InHomologicalAlgebra
I’d like to say that $RCh_*(F)$ is a point-set right derived functor of $Ch_*(F)$ but I don’t know how one knows that there is a fibrant replacement functor on $Ch_*(A)$ (here meaning the category of non-negatively graded cochain complexes), assuming $A$ has enough injectives.
Naively of course, one can construct an injective resolution $I_*(A)$ of an object of $A$. Given a map $A \to B$ one can build a map $I_*(A) \to I_*(B)$ but this is only unique up to chain homotopy equivalence and so this construction is not functorial on the point-set level (though of course suffices for defining total derived functors). Presumably this naive construction can be generalized to construct, given a non-negative graded cochain complex $A_*$, a cochain complex $I_*(A_*)$ of injectives together with a quasi-isomorphism $A_* \to I_*(A_*)$ but will similarly fail to be strictly functorial.
In the dual situation (non-negatively graded chain complexes) if we take the abelian category $A$ to be modules over some ring, then the appropriate model structure is cofibrantly generated so it’s clear there is a functorial cofibrant replacement. But I don’t know that the injective model structure is cofibrantly generated, even in this case.
Just a quick comment: in section 5.4 of
are some results on model structures on chain complexes that are provably not cofibrantly generated.
First, an apology: I accidentally posted this in the wrong place, under “MathForge general discussions.” I tried to delete it from there but couldn’t figure out how to do so. Could someone help me?
No apology necessary. I know it’s confusing to sometimes wind up there, but it is useful for me to have that place. I’ve deleted the original post.
Another technical tip: since you’re already using a Markdown formatting method, put ASCII angle brackets around a URI to make it live: http://ncatlab.org/nlab/show/derived+functor#InHomologicalAlgebra
In Hovey’s book Model Categories, Theorem 2.3.13 says that the injective model structure on chain complexes of modules over a ring is cofibrantly generated.
For an arbitrary abelian category with enough injectives, I sure don’t see any way to obtain a fibrant replacement functor. However, I’d expect that in many cases arising in practice (e.g. abelian sheaves), you could find a concrete injective-hull functor and thereby also a fibrant replacement functor, without necessarily needing to go through the machinery of cofibrant generation.
Thanks for these references and formatting tips. And for confirming what I’d suspected. Happy new year!
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