It is well known that for ordinary categories, if $C$ has finite limits and $D$ is cocomplete, and $A:C \to D $ is left-exact (i.e. preserves finite limits) then the left-Kan extension of $F$ along the Yoneda embedding $y:C \hookrightarrow Set^{C^{op}}$ is left-exact. I'm pretty sure this is still true for $\left(\infty,1\right)$-categories, once we replace the role of presheaves with that of $\infty$-presheaves, but is this written up somewhere?

  • $\begingroup$ Allow me to criticize myself here. This is well known for $D$ a TOPOS. If $D$ is not a topos, is it even true? $\endgroup$ – David Carchedi Dec 14 '11 at 1:39
  • $\begingroup$ For reference, let me record that this is false unless $D$ is a topos or something like it. For instance, take $C=1$, so that $F:C\to D$ picks out the terminal object. Then since coproducts in $\mathrm{Set}$ are disjoint, if the left-Kan extension of $F$ is left-exact then coproducts of copies of the terminal object in $D$ are also disjoint. This is false if, for instance, $D$ is a complete lattice. $\endgroup$ – Mike Shulman Apr 4 '17 at 7:57

For reference, at least when $D$ is an infinity topos, which I believe is probably necessary, this is Proposition in HTT.

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