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Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the inclusion $\operatorname{Kan} \hookrightarrow \operatorname{Cat}_\infty$. The following question came up in a discussion:

Suppose $D$ is an inversely-directed poset, and suppose $F:D\to \operatorname{Cat}_\infty$ is an injectively fibrant diagram of $\infty$-categories (wrt the Joyal model structure). Then is it true that the canonical map $\operatorname{Ex}^\infty(\lim F) \to \operatorname{lim}\operatorname{Ex}^\infty(F)$ is a weak homotopy equivalence (perhaps allowing for all of the limits in question to be appropriately homotopical)?

If not, is it true if $F(d)$ is isomorphic to the nerve of a poset for all $d\in D$?

Improve this question
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  • $\begingroup$ If you allow "for all of the limits in question to be appropriately homotopical", then your statement becomes tautologically true since the underlying ∞-functor of Ex^∞ is a right adjoint functor of ∞-categories. For the strict version, injectively fibrant towers can be characterized by the condition that all objects are fibrant and all maps are fibrations. So if you know that Ex^∞ applied to your maps yields Kan fibrations, then your map is a weak equivalence. $\endgroup$
    – Dmitri Pavlov
    Mar 22, 2019 at 16:36
  • $\begingroup$ @DmitriPavlov I'm pretty sure that Ex^∞ is a left adjoint, not a right adjoint. In particular, if G is an ∞-groupoid and ι: Kan -> Cat_∞ is the inclusion, then for any ∞-category X, Map_{Cat_∞}(X,ιG)~Map_{Kan}(Ex^∞ X, G). If Ex^∞ is also a right adjoint in the ∞-categorical sense, this is news to me. On-the-nose, it definitely doesn't preserve arbitrary cofiltered limits of simplicial sets. $\endgroup$
    – Steve
    Mar 22, 2019 at 17:20
  • $\begingroup$ Yes, I confused the directions of all arrows in my answer, so the claim was actually meant for direct limits, not inverse limits. Still, the strict version works just fine if you know that Ex^∞ applied to maps in your tower produces Kan fibrations. $\endgroup$
    – Dmitri Pavlov
    Mar 22, 2019 at 19:54
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    $\begingroup$ If this where true, this would imply that $Ex^\infty$ commutes with countable products of $\infty$-categories up to weak equivalences (seen as directed limit of finite products). An explicit counter example with a countable product of nerves of finite directed catégorie may be found in the introduction of Thomason’s paper « Cat as a closed model category ». $\endgroup$
    – D.-C. Cisinski
    Mar 23, 2019 at 9:46
  • $\begingroup$ @Denis-CharlesCisinski Thanks, perfect! $\endgroup$
    – Steve
    Mar 23, 2019 at 16:56

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