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So I was reading through Goerss-Jardine, and I realized that Lurie defines a model category in the HTT appendix to be a closed model category. To check this, I looked it up in Quillen's Homotopical Algebra, and indeed, Lurie requires that all model categories are closed. The nLab does this as well at model category.
If we use the standard definition of a not-necessarily-closed model category, h(C) is not the Bousfield localization, it is the quotient by the relation of left-right homotopy (using cylinder and path objects). If we assume all model categories are closed, the standard model structure on topological spaces is not a model structure at all, since weak equivalences are not always homotopy equivalences.
Has there been a change in terminology, or is there a reason why Lurie made this decision?
Edit: I found a discussion in Hovey. We should probably add something to the nLab page.
I don't recall what exactly Hovey writes. But I think it is common current practice to not say "Quillen closed model category" but just model category.
First of all this is the case that matters. Second the "closed" very badly collides with the notion of closed category.
But, yes, we should at least add a remark on terminology. I don't have the time. Would very much appreciate if you add something.
I will do that, but I'm just not sure that my statement about the model structure on Top is correct.
Edit: I know for a fact that the model structure on CGWH is closed, though.
Quillen's original word "closed" has been dropped by basically all workers in model category theory nowadays, having realized that "non-closed" model categories are not very common or useful. I don't know what you mean about topological spaces; Quillen's "closedness" condition was about cofibrations and fibrations determining each other by lifting properties; I don't think it says anything about homotopy equivalences.
I could have sworn that I read in one of the 6 sources I looked through that closedness => weak equivalences are homotopy equivalences, but I may have misread it (this is probable, since I was not reading the sources, but rather "finding in page" and scrolling through the results).
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