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If you take a look at the section Global Definition under homotopy limit, it says that we can define the homotopy limit when we only have a fixed class of weak equivalences instead of a model structure. It then refers the reader to homotopy Kan extension, which only discusses this in terms of simplicially enriched monoidal model categories.
How can we define the homotopy limit in the way it says we can at homotopy limit using only weak equivalences?
have a closer look at the entry: you’ll see that homotopy Kan extension defines homotopy limits (also) in any Kan-complex enriched category!
You get these from a category with weak equivalences from Dwyer-Kan localization.
Is this covered in Dwyer-Hirschhorn-Kan-Smith? Also, is there a way to do it without Dwyer-Kan localization?
Is this covered in Dwyer-Hirschhorn-Kan-Smith?
Not sure, I guess so. In any case, our nLab entry on Dwyer-Kan simplicial localization needs to find somebody to add more references and details…
Also, is there a way to do it without Dwyer-Kan localization?
Well, all you need is the left/right derived functor of the limit and colimit functors. But it can be hard to get your hands on these without any further tools. Recall that derivators are another way to talk about homotopy limits without using model structures or simplicial enrichment.
As Urs said, you can define these things to be derived functors, which make sense as soon as you have weak equivalences; but usually in order to show that they exist or say anything interesting about them, you need more. The language of derivators doesn’t really change that, since in order to produce a derivator, you usually need to have a model structure or an enrichment or something.
@Mike: I read the page over at derived functor. Could you explain why having non-pointwise Kan extensions is not good?
Also, I’m reading a paper by Barwick and Kan about the homotopy theory of “relative categories”, i.e. pairs consisting of a category C and a lluf subcategory W, which they prove is Quillen equivalent to the homotopy theory of quasicategories, simplicially-enriched categories, etc. From this result, shouldn’t it follow that we can give a good notion of a homotopy Kan extension in this case (which appears to be the same as “categories with weak equivalences”) without requiring all of those other structural equipments?
Many of the good properties of Kan extensions depend on their being pointwise; for instance, there are a bunch of these considered in Kelly’s book on enriched category theory.
shouldn’t it follow that we can give a good notion of a homotopy Kan extension in this case (which appears to be the same as “categories with weak equivalences”) without requiring all of those other structural equipments?
I don’t know – how are you thinking would it follow? And what do you mean by “good”? As I said, you can define what a derived functor would mean in the generality of categories with weak equivalences, but they may not exist.
(I think Barwick and Kan’s “relative categories” are almost the same as categories with weak equivalences, but they don’t require the 2-out-of-3 property. Note in particular that the morphisms in their category of relative categories preserve weak equivalences, but the nontrivial question of derived functors is what to do about functors that don’t preserve weak equivalences.)
By the way we have an entry on Barwick-Kan: model structure on categories with weak equivalences.
(This is not to say that it answers Harry’s question – I am not sure what exactly the question is now! – but just to remind us that for whatever information we mention here, there are precisely two options: either it’s already in that entry. Or else, somebody should put it in that entry! )
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