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I'd be very happy if the question When do Kan extensions preserve limits/colimits? has been fully answered. But it seems not.

I have a more specific question though. Let $C$ be a site (essentially small to be safe) equipped with some subcanonical Grothendieck topology, $PSh$ be its category of presheaves. Then it's well-known that for any functor $C \to B$ with $B$ cocomplete, the left Kan extension $PSh \to B$ along the Yoneda embedding $C \to PSh$ necessarily preserves colimit. Is it true if I replace $PSh$ by the category of sheaves $Sh$? In other words, does the left Kan extension along $C \xrightarrow{Yoneda} PSh \xrightarrow{sheafification} Sh$ preserve colimit? If in general not, is there any sufficient condition we can impose on the functor $C \to B$?

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Usually it does not. We can take $C \to B$ to be the universal example of a functor to a cocomplete category, namely the Yoneda embedding $C \to PSh$. Then the left Kan extension of $y : C \to PSh$ along $j : C \to PSh \to Sh$ is the inclusion $Sh \to PSh$, and this usually does not preserve colimits. The calculation of this left Kan extension can be verified using the coend formula: $$ (Lan_j y)(X) = \int^{c : C} Sh(jc, X) \otimes yc = \int^{c : C} X(c) \otimes yc = X $$ where the second equality used the fact that the topology is subcanonical.

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  • $\begingroup$ Thanks for the counterexample! Is there any sufficient condition on the functor $C \to B$ that can guarantee the left Kan extension is colimit preserving? $\endgroup$ – gregodom Jun 10 '20 at 14:08
  • $\begingroup$ E.g. if $F:C \to B$ is a subfunctor of a functor $F':C \to B$ whose left Kan extension is colimit preserving, is the left kan extension of $F$ colimit preserving? $\endgroup$ – gregodom Jun 10 '20 at 14:45

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