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Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a functor. Denote $G$ the left Kan extension of $F$. Then, is $G$ continuous, i.e. does it commute with colimits?

Does this extend to the world of $\infty$-categories? Namely, let $\mathcal{C}$ be an $\infty$-category and $h_-: \mathcal{C} \to (\infty-\mathrm{groupoids})^{\mathcal{C}^{op}}$ be the Yoneda embbeding. Let $\mathcal{A}$ be a cocomplete $\infty$-category, $F: \mathcal{C}\to \mathcal{A}$ a functor and $G$ its left Kan extension. Then, we can also ask if $G$ is continuous. Here, colimit is of course understood to be in the appropriate sense, i.e. homotopy colimit.

Thank you.

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    $\begingroup$ See ncatlab.org/nlab/show/free+cocompletion and ncatlab.org/nlab/show/…. $\endgroup$ – Todd Trimble Apr 25 '14 at 3:00
  • $\begingroup$ @ToddTrimble: That's very nice! I didn't think that there is such a readily available adjoint! $\endgroup$ – QcH Apr 25 '14 at 3:28
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    $\begingroup$ Is there some reason that you call a functor preserving colimits "continuous"? I usually call this "cocontinuous". The nlab agrees. $\endgroup$ – Tim Campion Apr 29 '14 at 23:03
  • $\begingroup$ @TimCampion: No idea, I'm just following Gaitsgory's terminology. $\endgroup$ – QcH Apr 30 '14 at 4:56

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