## relationships between properties of model categories

I've recently found myself running up against all sorts of adjectives that can describe a model category: cofibrantly generated, combinatorial, tractable, stable, locally (finitely) presentable, (left and/or right) proper, simplicial, admits Bousfield localizations, et al. I'm hoping for a reference whose explicit goal is to give an intuitive explanation of these concepts; presumably, such a reference would also tell me what each of these adjectives can buy me.

(As a bonus, this reference might even have a diagram analogous to the one on pp. 576-7 of Gortz & Wedhorn's book Algebraic Geometry, which summarizes the relationships between different properties that a morphism of schemes may satisfy. But that'd just be for fun.)

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 In the case of algebraic geometry there's something to prove for many of the indicated relations. For model categories, and for the properties you mention, one only has to look at the definition and see whether one property is required in the definition of another property. This is why I think this question is somewhat empty as currently stated. – Fernando Muro Sep 13 at 0:52 I've never come across such a list, and I agree with Fernando that it wouldn't be very useful for the properties that you've listed. However, a few things are known. If you add "admits left Bousfield Localizations" to your list then left proper and combinatorial implies this. So does left proper and cellular. If you replace simplicial by "quillen equivalent to a simplicial model category" then Dan Dugger has a paper saying which of the other properties imply this: pages.uoregon.edu/ddugger/smod.html. There's also "permits small object argument" $\Rightarrow$ cofibrantly generated. – David White Sep 13 at 1:17 Thanks for the feedback. The reason I asked is that I've seen these sorts of adjectives running around before, but only recently have I begun to actually wrestle with them. I wasn't sure a diagram would be helpful, but the one in Gortz-Wedhorn popped into my head when I was thinking about it. I suppose the more important part is the last sentence. Certainly nLab has been a good resource, but I think I'd benefit from something a little more narrative. I'll edit. – Aaron Mazel-Gee Sep 13 at 11:33 Well, the classical reference is Hovey, but he doesn't cover a lot of the adjectives on your list. Hirschhorn covers more and is very thorough, so that's where I would (and did) start. To my knowledge those are the only books on model categories, though there are large sections of Lurie's DAG which cover the subject and you could do a CTRL+F search on those for the adjectives of interest (esp tractable, finitely presentable, etc). See also Chorny's recent work on class-combinatorial & class accessible categories. Otherwise I recommend Dugger. His writing is very clear and down to earth. – David White Sep 13 at 13:52 Ah, so there's no magic pill that I can take and suddenly understand all this stuff? ;o) I'll give this another day or so, and if nobody else responds then I'll accept your comment as an answer. Already Hirschhorn's introduction (as well as his introductions to the various Parts and Chapters) has been quite helpful. – Aaron Mazel-Gee Sep 14 at 22:08
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