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I was wondering if someone could explain some of the concrete applications of model categories. My possibly naive understanding of the motivation is that one wants to mimic the category of topological spaces in some sense or to define a homotopy theory for a category.

For example, more concretely on the Wikipedia page it is stated that the category of chain complexes of $R$-modules for some commutative ring $R$ is a model category and that homology can be viewed as a type of homotopy which allows generalisation of homology to objects such as groups and $R$-algebras. Is there some reference where there is explained a bit more? How do model categories allow one to generalise homology?

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    $\begingroup$ Could you focus the question? "[What are] concrete applications of model categories" and "How do model categories allow one to generalise homology" seem to go in different directions... $\endgroup$ – YCor May 24 '20 at 2:47
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    $\begingroup$ Arguably, the primary concrete application is to model all sorts of computations with the underlying (∞,1)-category by performing computations with the underlying 1-category. This includes deriving functors of all sorts, in particular, limits and colimits, monoidal product and internal hom, etc. This is explained in detail here: ncatlab.org/nlab/show/homotopy+theory+FAQ $\endgroup$ – Dmitri Pavlov May 24 '20 at 5:39
  • $\begingroup$ "one wants to mimic the category of topological spaces in some sense or to define a homotopy theory for a category": I am not sure what the mimicking process refers to. One does not need model categories to define homotopy categories, though, since merely having weak equivalences is already sufficient. $\endgroup$ – Dmitri Pavlov May 24 '20 at 5:41
  • $\begingroup$ Hi Dmitri, yes this looks like what I had in mind, thanks for this. $\endgroup$ – Hollis Williams May 24 '20 at 12:31
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There are many references where model categories, and their connection to homology, are described more. See this MO question for a list. For the example of $Ch(R)$, there are several model structures. Those that have quasi-isomorphisms as weak equivalences capture homological algebra (e.g., a morphism $f_*: C_* \to D_*$ is a weak equivalence if the map on homology $H_*(f)$ is an isomorphism). Similarly, there are model structures on the category of topological spaces, either for classical homotopy theory (where $f$ is a weak equivalence if all $\pi_*(f)$ are isomorphisms) or for homology (where $f$ is a weak equivalence if all $H_*(f)$ are isomorphisms). Indeed, this can be done for generalized homology theories, and the machinery of Bousfield localization allows you to change your model structure to focus on an enlarged class of weak equivalences in this way.

When the wikipedia page mentioned groups and $R$-algebras, it probably had in mind Quillen's early work producing model structures for simplicial groups and simplicial $R$-algebras. In these cases, the homotopy theory is lifted from the category of simplicial sets (a model for topological spaces), and the Dold-Kan correspondence gives an equivalence $Ch_{\geq 0}(\mathcal{A}) \simeq Fun(\Delta^{op},\mathcal{A})$ for an Abelian category $\mathcal{A}$. This is very clearly described in a note by Akhil Mathew. The homotopy theory of homological algebra on one side is equivalent to the homotopy theory coming from spaces on the other side, justifying the sentence on wikipedia.

There are also model structures on the category of groups (e.g., the trivial model structures), but they are boring. There are model structures on $R$-algebras for entirely different homotopy theories related to representation theory, e.g., you can start with the model structure on $R$-modules whose homotopy category is the stable module category (described in Hovey's book among other places), and can lift it to a model structure on $R$-algebras, as I wrote about here. Homotopy in these cases is about having the same representation theory in the stable module category (i.e., after quotienting out boring representations). This is probably not what wikipedia had in mind, but provides lots of applications of model categories.

There's far too much to possibly fit in a single MO answer, so I encourage you to read the references linked above, and ask more targeted questions as you go. Model categories have lots of applications, going far beyond "just" modeling computations in the underlying $(\infty,1)$-category (though, surely, this is an important application of the theory). For instance, when Javier Gutierrez and I proved the Blumberg-Hill conjecture using model categories, this really required the cofibrations, rather than only the weak equivalences. I don't know how to carry out that proof on the level of $(\infty,1)$-categories. The same is true of my recent work with Michael Batanin proving a generalized form of the Baez-Dolan Stabilization Hypothesis.

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  • $\begingroup$ Hi David, thanks for this. I'm wondering for example, one can have homological representations of certain types of group (for example, braid groups act on the homology of topological spaces obtained from punctured discs via functor constructions), might it be possible to use model categories to generalise these constructions and obtain new representations for those groups? $\endgroup$ – Hollis Williams May 24 '20 at 16:13
  • $\begingroup$ That's an interesting question. I would need to think carefully about it. My first instinct was "no: model categories are for organizing information, not obtaining new representations" but actually I can imagine a framework where you could create new representations via categorical procedures. I'm curious to think about encoding braid group actions on homology in an abstract framework. I don't have enough time now to think about this (textbook deadline soon), but maybe in a few weeks. $\endgroup$ – David White May 24 '20 at 17:28
  • $\begingroup$ In your previous comment you mentioned categorical procedures, do you mean categorical methods as compared to model categories? $\endgroup$ – Hollis Williams Aug 30 '20 at 19:50
  • $\begingroup$ I guess both, but of course you only need the model structure for the latter. Like, you could create new representations from old via pushouts or other colimits (categorical) or via cofibrant replacement (model category technique). All of this is speculative since we don't actually have a specific model structure in mind. $\endgroup$ – David White Aug 30 '20 at 23:03

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