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I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm equally happy to work with quasicategories. Let $S(W_n)$ denote the saturation of $W_n$ in $\mathcal{M}$ -- i.e. those morphisms which become weak equivalences when I invert $W_n$. I would like to calculate $\bigcap_n S(W_n)$. Can anyone suggest any tools for doing this? I would be very happy to wind up with some explicit morphisms $W_\infty$, built from morphisms in the $W_n$, such that $S(W_\infty)=\bigcap_n S(W_n)$.

Edit: per Dmitri's comment, I describe the $W_n$ which interest me. However, my particular interest notwithstanding, I really would like to learn more about tools with which one might attack such questions in general. My intent in asking my question was not merely to have someone solve my problem for me.

Here are my observations so far. $S(W_n)$ is the class of morphisms which become equivalences upon homming into the objects of $D_n$, the $W_n$-local objects. So $W_\infty\stackrel{def}{=}\bigcap_n S(W_n)$ will be the morphisms which have this property for all objects in $\bigcup_n D_n$. This will still hold for any limits of objects in $\bigcup_n D_n$. Unfortunately, this criterion feels more useful for establishing when a morphism is not in $W_\infty$.

A more useful observation is that colimits are left adjoints and hence commute with localization (here I should ideally be working in a quasicategory so that I don't have to deal with sticky technicalities about cofibrations, but even if I am still in the model category world, there is probably some cofibrant replacement I can take). This means that if I can construct a natural transformation $\phi:F\to F'$ of diagrams $F,F':\mathcal{C}\to\mathcal{M}$ such that for each $n$, $\phi$ is an objectwise equivalence on some final set of objects in $\mathcal{C}$, then the induced morphism of colimits will be in $W_\infty$ even if none of the objectwise components of $\phi$ are. I think I may be able to get somewhere with this observation, and so maybe my question has become somewhat unneeded. But if anyone can suggest other/further tools/strategies, I will still be grateful.

Given an object $\theta\in\Theta_n$ where $\Theta_n$ is Joyal's cell category, consider $\theta$ as a strict $n$-category. We have a map $\theta\to\theta'$ which collapses each sequence of parallel $n-1$ morphisms. $\theta'$ is also an object of $\Theta_n$; I only invoke the $n$-categorical terminology to help me explain the maps I mean. Let $T_n$ be the collection of all such maps in $\Theta_n$, but consider the $T_n$ as a collection of maps in $\Theta_\infty$, the union (colimit) of the $\Theta_n$.

Consider the morphisms in $s\mathcal{S}et^{\Theta_\infty^{op}}$ represented by the morphisms in $T_n$. By abuse of notation, denote these classes of representable morphisms $T_n$ as well. Then $W_n\stackrel{def}{=}\bigcup_{m>n}T_m$.

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  • $\begingroup$ I think it would be extremely helpful if you explained what specific choices of M and W_k you are interested in. $\endgroup$ – Dmitri Pavlov Jun 1 '18 at 15:59
  • $\begingroup$ Sure. I will add an edit. $\endgroup$ – Kyle Ferendo Jun 2 '18 at 1:14
  • $\begingroup$ Basic question: is there any morphism which is not an equivalence which you can show is in $\cap_n S(W_n)$? $\endgroup$ – Tim Campion Jun 3 '18 at 17:03
  • $\begingroup$ @TimCampion, very good question! I'm about certain the answer is yes. I can describe a morphism that should be in $\bigcap_n S(W_n)$, and I have a sketch of a proof, but making that proof fully rigorous will involve working through a few technicalities (and we know this has gotten me into trouble in the past!). I'm feeling pretty good now: I can describe (with proof) the local objects, and I believe I can give an example of a non-equivalence that gets inverted. The only thing that would be icing on the cake would be a proof that this morphism generates the rest. $\endgroup$ – Kyle Ferendo Jun 8 '18 at 15:02

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