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What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?

Stable ∞-categories do satisfy this property, but are there any others?

(The application I have in mind involves a sheaf on the site of smooth manifolds valued in some ∞-category, the finite limit is the descent object for some finite cover, and the colimit over Δ^op is the realization of a simplicial object obtained from the sheaf.)

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    $\begingroup$ Any $\infty$-category $\mathcal{C}$ which admits a conservative, finite-limit preserving, and sifted colimit-preserving functor to a stable $\infty$-category will have the same property. For example, all sorts of structured ring spectra ($E_{\infty}$, $A_{\infty}$, etcetera...) $\endgroup$
    – Jacob Lurie
    Commented Jul 26, 2016 at 23:57
  • $\begingroup$ @JacobLurie: Thanks for the example! I guess this applies in general to algebras over monads in stable ∞-categories whose functor preserves sifted colimits. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 27, 2016 at 6:13

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