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It is well known that for a functor $F: \mathcal R \to \mathcal M$, where $\mathcal R$ is a Reedy category and $\mathcal M$ is suitably bicomplete, the following decomposition determines the structure around $y \in \mathcal R$: $$ Latch(F)(y) \to F(y) \to Match(F)(y); $$ here we decompose the canonical map between the latching and the matching objects at $y$.

Is there a reference with a written down proof for the same phenomenon occurring in higher categories? Namely, an extension theorem saying that the category of functors from $\mathcal R$ to a bicomplete higher category has the same inductive property, that is, to reconstruct a functor on objects of degree $k$ all we need is to factor the latching-to-matching maps constructed using objects of degree $\leq k-1$?

Thank you in advance.

EDIT: following comments, what interests me is the $(\infty,1)$ case where no presentability is assumed, hence a strictification technique cannot be applied easily.

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  • $\begingroup$ An $(n,k)$ category $\mathcal{M}$ is in particular an $(n,1)$-category by discarding noninvertible higher morphisms. Latching and matching are certain (co)limits, and thus depend only on the $(n,1)$-structure of $\mathcal{M}$. So it suffices to show this when $\mathcal{M}$ is an $(\infty,1)$-category. Then if $\mathcal{M}$ is presented as a model category, this is just the usual theory of the Reedy model structure. Are you particularly interested in cases where $\mathcal{M}$ is not presented as a model category? $\endgroup$
    – Tim Campion
    Commented Nov 10, 2017 at 14:22
  • $\begingroup$ Yes, will edit. $\endgroup$
    – Edouard
    Commented Nov 10, 2017 at 15:03

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This follows from Proposition A.2.9.14 in Higher Topos Theory. Lurie expresses a Reedy category as inductively obtained by homotopy pushouts in the Joyal model structure, and using this one gets the description you're after. He proves this explicitly for ordinary categories in A.2.9.15 and remarks in A.2.9.16 that the same proof extends to diagrams in $\infty$-categories.

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