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a few more details at model structure for left fibrations
The page model structure for left fibrations claims that this model structure is proper. However, the cited propositions in HTT only claim that it is left proper. Is it right proper? If so, where is a proof?
Thanks for catching this. I have fixed it. Not sure why the “left” got missing.
Suppose $X$ is a quasicategory. The model structure for left fibrations on $sSet/X$ is a left Bousfield localization of the model structure on an over category arising from the model structure for quasi-categories, since it has the same cofibrations and fewer fibrant objects. Can one exhibit a nice small set of maps at which it is the localization?
Well, the left model structure on $\mathbf{sSet}_{/ X}$ is enriched with respect to the Kan–Quillen model structure on $\mathbf{sSet}$, and all objects are cofibrant, so the following is true:
(I am abusing notation by omitting the projections to $X$.) On the other hand, given a set $\mathcal{S}$ of morphisms in $\mathbf{sSet}_{/ X}$, since the slice Joyal model structure is enriched with respect to the Joyal model structure on $\mathbf{sSet}$, the following should be true:
Putting these two together, it would seem to me that to get the desired $\mathcal{S}$, it is enough to check that the $\mathcal{S}$-local objects you get are the left fibrations. So I think $\mathcal{S} = \{ \{ 0 \} \hookrightarrow \Delta^n : n \ge 0 \}$ works.
Hmm, when you have an enriched model structure I think you actually get two different notions of locality depending on whether you use the enriched hom-objects or the model-categorical hom-spaces, and the latter is the one that corresponds to the usual sort of Bousfield localization. So I don’t believe your second bullet; instead I would expect to see $f^*$ acting on maximal sub-Kan-complexes of those mapping spaces.
Secondly, how do you know that your $\mathcal{S}$ (which is the obvious choice) does in fact detect the left fibrations?
Hmmm, maybe the $\mathcal{S}$ I suggested doesn’t quite work. How about $\mathcal{S} = \{ \Lambda^n_0 \hookrightarrow \Delta^n : n \gt 0 \}$ instead?
Okay. And I think your argument works for ordinary localization too, replacing the internal hom-objects with their maximal subgroupoids and “isofibration” with Kan fibration.
Pedro Boavida has just pointed out to me that your $\mathcal{S}$ in #7 is also the answer asserted by Moerdijk-Heuts on page 5 here.
By the way, I assumed you meant the set of all maps of the form $\Lambda^n_0 \hookrightarrow \Delta^n$ over $X$, i.e. indexed not just by $n$ but by a map $\Delta^n\to X$.
I have added a mention of this fact to the page model structure for left fibrations.
Re #9: Yes, of course. One of the irritations of this abuse of notation.
The definition of weak equivalence appearing on the page (and in HTT) is rather mysterious. (It is much too clever for a definition!) I think I like this Bousfield localisation definition better.
But can you show that the fibrant objects in the Bousfield localization are precisely the left fibrations without already knowing that the model structure as defined by Lurie exists?
Well, let’s see. We know every $\mathcal{S}$-local object is a left fibration. Conversely, an isofibration $C \to S$ is $\mathcal{S}$-local in the Joyal-enriched sense if and only if it has the right lifting property with respect to $\Delta^m \times \Lambda^n_0 \cup \partial \Delta^m \times \Delta^n \hookrightarrow \Delta^m \times \Delta^n$. But that inclusion is a left anodyne extension by [HTT, Corollary 2.1.2.7], so we are done.
Every time I say “Bousfield localization”, I mean in the ordinary sense, not the Joyal-enriched sense.
It’s obvious that $\mathcal{S}$-local in the Joyal-enriched sense implies $\mathcal{S}$-local in the ordinary sense (for fibrant objects).
Okay, fair enough. Although HTT 2.1.2.7 is already more than half the way towards Lurie’s construction of the left fibration model structure.
Sure. That’s also a formal consequence of the claim that the left fibration model structure is a left Bousfield localisation of the sliced Joyal model structure.
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