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I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ groups for some Hopf algebroid.

Given an abelian category $\mathscr{A}$ with a category of chain complexes $Ch(\mathscr{A})$, the homotopy category $K(\mathscr{A})$ is defined as the naive homotopy category (i.e. replace chain maps with chain homotopy classes of chain maps) but with quasi-isomorphisms inverted (these are the maps which induce isomorphisms on homology). This is obviously the result of placing some kind of model structure on $Ch(\mathscr{A})$. This leads me to consider a few obvious questions.

  1. Are there alternate, interesting model structures for $Ch(\mathscr{A})$?
  2. (How) has the advancement of model category theory impacted the study of things such as chain complexes, and perverse sheaves for example?
  3. Are there any good treatments of homological algebra which makes use of the rich theory of model categories?
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    $\begingroup$ Note that there are in fact two model structures that are commonly used - the injective and projective model structures, which correspond to taking injective and projective resolutions respectively. (At least naively, each of these require different boundedness conditions on the chain complexes because we resolve in different directions. Putting a model structure on unbounded chain complexes is trickier. This should not be surprising because we cannot inductively construct injective/projective resolutions for unbounded chain complexes) $\endgroup$
    – Dexter Chua
    Commented May 8, 2020 at 9:19
  • $\begingroup$ @DexterChua of course! When I say "alternate" model structures I certainly mean alternate to (at least) these two model structures. Could you say anything about the unbounded case? For example I'm guessing it still has weak equivalences the quasi-isomorphisms and I would certainly hope that it is cofibrantly generated $\endgroup$
    – asd
    Commented May 8, 2020 at 11:26

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I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: http://nyjm.albany.edu/j/2014/20-53p.pdf. Its title is Six model structures for DG-modules over DGAs: Model category theory in homological action. The theme is how different model structures can illuminate concrete calculations. I had computed the cohomology of various homogeneous spaces, way back in the 1960's, using some strange looking explicit cochain complexes. Tobi Barthel, Emily Riehl and I found that those turn out to be explicit examples of a variant kind of cofibrant approximation.

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Yes, there are zillions of model structures on Ch(A), corresponding to whatever class of projectives you choose to use for your homological algebra. This is all spelled out in the paper "Quillen model structures for relative homological algebra" by Christensen and Hovey.

More generally, the theory of cotorsion pairs builds model structures in many algebraic settings (including quasi-coherent sheaves; I'm not sure about perverse sheaves, not even if they satisfy bicompleteness as a category). Hovey's seminal paper is here. A great survey by Gillespie is here. This material has also appeared in books. A recent one by Marco Perez is here.

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