When answering this question,

In a model category, is the full subcategory of fibrant objects a reflective subcategory?

I realized that I wasn't even sure what the correct definition of a model category was! When I first learned about the basics of model category theory, I learned from Mark Hovey's book "Model categories". Mark Hovey defines a model category to be

Definition 1.1.3. A model structure on a category C is three subcategories of C called weak equivalences, cofibrations, and fibrations, and two functorial factorizations (α, β) and (γ, δ) satisfying the following properties:

Hovey then goes onto describe the usual properties of a model category that we've all come to know and love. The interesting part here is that Hovey requires the factorization systems to be functorial. This means that one could have two different model structures on a category even if they have the same factorizing subcategories.

On the other hand, I have generally been told (and accepted) that model categories ought to be equipped with weak factorization systems, which does not require functoriality. I personally think this definition, which disagrees with Hovey's, is better suited for the underlying philosophy of homotopy theory, but I won't go into that.

Question: Is there a generally accepted concensus on what a model category is? Can anyone think of any propositions or theorems in Hovey's book, whose proofs rely heavily on functoriality of the factorization?

  • $\begingroup$ He discusses the choice of the functorial factorization in the book, as I recall, and uses it constantly, for instance in getting to his main results about every homotopy category being a module over the classical homotopy category. $\endgroup$ – Kevin Carlson Feb 7 '16 at 5:27
  • $\begingroup$ @Kevin Carlson Yes, I think he says that in all the examples he considers, the factorization can be made functorial. I also know he uses the fact frequently. I don't think a weak factorization system on a model category always implies the existence of a functorial factorization. I'm looking for examples where statements cannot be proved without using functoriality. $\endgroup$ – user113529 Feb 7 '16 at 5:50
  • $\begingroup$ Isn't there also some condition about having finite / all limits / colimits? $\endgroup$ – Qiaochu Yuan Feb 7 '16 at 6:55
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    $\begingroup$ @QiaochuYuan I just looked it up. Hovey apparently distinguishes between a category with a Model structure and a Model category. The latter is a category having all small limits/colimits, equipped with a Model structure. $\endgroup$ – user113529 Feb 7 '16 at 7:13
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    $\begingroup$ I'm not aware of any results that genuinely require functorial factorisation – however, doing without is sometimes a lot harder. See also here and here. $\endgroup$ – Zhen Lin Feb 7 '16 at 8:22

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