# Left Kan extension that preserves colimit

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 $$C \to B$$?

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.
• 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? – gregodom Jun 10 '20 at 14:08
• 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? – gregodom Jun 10 '20 at 14:45