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started adding some genuine substance to model structure on sSet-categories (which used to be just a template).
the original definition of the fibrations at model structure on sSet-categories was a bit ambiguous. I have replaced it by the more useful formulation: (degreewise Kan fibration and) isofibration on homotopy categories.
Is the definition of Dwyer-Kan weak equivalence at model structure on sSet-categories correct? I note that Julie Bergner in her article, (TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 5, May 2007, Pages 2043–2058 ) uses a different set of conditions with use of $\pi_0\mathcal{C}$ instead of the ‘homotopy category’. That latter is a bit suspect or confusing anyhow because until we have a class of weak equivalences we cannot form a homotopy category in the strict sense of homotopy category, (well, that possibility is almost handled there but it is not explicit and so is a bit confusing). I looked for the entry that describes the category given by applying $\pi_0$ to each $\mathcal{C}(x,y)$, but could not remember what it was called here.
Once I know that I can fix the entry, but I want to check that I haven’t missed some point!
It seems to be conventional to call the $\pi_0$ category the ‘homotopy category’ – this is what Lurie [Higher topos theory] and Rezk [A model for the homotopy theory of homotopy theories] also do, for example. Lawvere [Axiomatic cohesion] seems to attribute this kind of construction to Hurewicz.
It’s as Zhen Lin says. But it is true that it wouldn’t hurt to clarify this in the entry.
I would say it is rather the page homotopy category that needs correction if its definition does not include this. Isn’t passing to $\pi_0$ of hom-spaces the original meaning of “homotopy category”?
Mike, your comment is exactly my thought, and I had a sneaky feeling that it was homotopy category that might need attention.
My own view is that that entry is in a bit of a mess! Harry’s comment is relevant here. It is a question of conventions I think. Classically and conceptually the homotopy category (of spaces) is formed by dividing out by the homotopy relation. It is then a theorem that if you take $spaces[homotopy equivalences^{-1}]$ you get the same thing. The $C[\Sigma^{-1}$ is a ’category of fractions’ not a ’homotopy category’ in such classics as Gabriel and Zisman, although it can be argued that that term makes more sense if there is a ’calculus of fractions’ around. (There is an old paper by Bauer and Dugundji that explores the relationship with homotopy a bit more.) … so which comes first for a modern nPOV treatment? The quasicat viewpoint could be emphasised and that could be firmly based in the original meaning.
It’s not a deep point either way, just a matter of language convention. Just add a remark for clarification where you see the need.
Tim, feel free — I don’t have time.
Hmmm. I’m inclined to agree with Tim Porter: it is a somewhat non-trivial theorem that the $\pi_0$ category is also a localisation in the sense of inverting weak equivalences. In fact it’s not even true for simplicial model categories unless one cuts down to the full subcategory of cofibrant–fibrant objects. In general, the best we can say is that the $\pi_0$ category of a simplicially tensored or cotensored category is the localisation at the simplicial homotopy equivalences. The situation for cohesive toposes appears to be even more murky.
it is a somewhat non-trivial theorem that the π 0 category is also a localisation in the sense of inverting weak equivalences.
That’s not being doubted. It’s just about what a word is to mean in some article. Just add a comment on the nLab page that clarifies it.
As I unconverted and unrepentant shape theorist, Urs, I beg to disagree. When you choose a QMC structure, or even a cat. with w.e. structure, you are choosing to filter out information from the start. (Of course, this is very useful, but really has little to do with homotopy!) If you want to study phenomena that are exhibited by non-cofibrant /non-fibrant objects then you are sunk as you have to replace them by a single replacement. (Of course, there are deeper ways around this, but that is a slightly different point.)
I will go and try to change the entry.
Tim, you cannot disagree with me, since I didn’t make a statement. I am just suggesting that you add a clarification to the page where you see the need.
Here is a statement:
It’s not a deep point either way, just a matter of language convention.
;-)
I have tried to clarify things a little. I also took out the old ’discussion’ between Harry and myself.
More seriously, that page could perhaps benefit from having something on ’homotopy category of quasi-categories’, or should we direct ’the reader to some other entry for that?
Reminder to myself: add the proposition that the model structure on sSetCat is right proper.
Is it left proper? The Dwyer–Kan model structure for simplicial categories over a fixed set of objects is both left proper and right proper, and they used left properness to justify their construction of the standard simplicial localisation.
re #15: done
re #16: dunno
added pointer to:
there was an old claim here, without proof or reference, that Quillen equivalences induce Dwyer-Kan equivalences.
I have added:
This is made explicit in Mazel-Gee 15, p. 17 to follow from Dwyer & Kan 80, Prop. 4.4 with 5.4.
also
The analogous statement under further passage to Joyal equivalences of quasi-categories is Lurie 09, Cor. A.3.1.12, under the additional assumptions that the model categories are simplicial, that every object of $C$ is cofibrant and that the right adjoint is an sSet enriched functor.
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