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Added a section with a little bit of detail on model structures on cochain complexes in non-negative degree to model structure on chain complexes.
I now finished adding a detailed proof of the model structure on non-negatively graded cochain complexes of abelian groups with fibrations the degreewise surjections (and weak equivalences the usual quasi-isos).
In this subsection.
The same proof should go through for non-negatively graded chain complexes.
The result seems to be standard or maybe folk lore(?), but I had trouble tracking down an explicit proof. Explicit standard proofs are widely available for the case that the ambient category has all objects injective, but I am not aware of one for the ambient category being . However, I wouldn’t be surprised at all to hear you say “But this is in Quillen’s age old article xyz!” or the like. If so, please do.
added a little bit of the central statements of Christensen-Hovey’s article to model structure on unboiunded chain complexes
added at model structure on chain complexes a paragraph (in a new Properties-section) on Quillen adjunctions from complxes of sheaves into simplicial sheaves. But did not get into the discussion of the passage to the local structure for the moment.
I am glad that unbounded complexes started to appear in nlab. Assuming that everything is bounded and in characteristics zero, hence equivalent to (co)simplicial picture was repeling many algebraic geometers for example from using nlab, and I am glad to see it is moving toward covering the wider traditional scope.
I had two small mistakes in the statement of the model structure in terms of -monomorphisms in the section general results. I think I have fixed things now.
I started a section Left/right exact functors and Quillen adjunctions but I have to dashh off now
I am glad that unbounded complexes started to appear in nlab.
Already since a few months!
Dmitry (Roytenberg) says there is at least one typo somewhere on model structure on chain complexes where it says “chain” where it must say “cochain” or conversely. I’ll try to sort this out with him tomorrow.
I have polished the organization of the section on complexes in non-negative degree a bit.
to model structure on chain complexes I have added after the statement of the injective Quillen model structure the remark
This means that a chain complex is a cofibrant object in the injective model structure precisely if it consists of projective modules. Accordingly, a cofibrant resolution in the injective model structure on chain complexes is precisely what in homological algebra is called a projective resolution.
This way the traditional definition of (left) derived functor in homological algebra relates to the general notion of left derived functor. See there for more details.
I thought I had similar remarks about the relation of model category theory to traditional homological algebra added long ago. Certainly I did in the Idea-sections, but maybe I never spelled out the details. Yet.
Surely you mean this paragraph to be about the projective model structure?
Thanks, fixed.
In the unbounded complexes section, I added a subsection on the approach of Gillespie using cotorsion pairs. In connection with that I also created
Soon I plan to also add a discussion on the third approach, by Cisinski-Deglise (which Urs has already added the reference to).
have finally given the projective model structure on UNbounded chain complexes its own section, citing the results by Hovey-Palmieri-Strickland, Schwede-Shipley and Rezk-Schwede-Shipley.
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