When do Kan extensions preserve limits/colimits?

Question

I'm guessing the answer to this question is well-known:

Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}_Y\left(F\right)$ preserve colimits? Notice that if $C=P$ and $Y=id_C,$ then $\mathbf{Lan}_Y\left(F\right)=F,$ so this is not true in general. Would $F$ preserving colimits imply this?

Dually, under what conditions does a right Kan extension preserve limits?

Thank you.

Answer 1

The pointwise left Kan extension of F along Y is a coend of functors $Lan_{Y}(F) = \int^{x}P(Yx,-).Fx$ where each functor $P(Yx,-).Fx$ is the composite of the representable $P(Yx,-):P \to Set$ and the copower functor $(-.Fx):Set \to D$. As a coend (colimit) of the $P(Yx,-).Fx$ the left Kan extension preserves any colimit by each of these functors.

Now the copower functor $(-.Fx)$ is left adjoint to the representable $D(Fx,-)$ and so preserves all colimits, so that $P(Yx,-).Fx$ preserves any colimit preserved by $P(Yx,-)$. Therefore $Lan_{Y}(F)$ preserves any colimit preserved by each representable $P(Yx,-):P \to Set$ for $x \in C$.

If Y is the Yoneda embedding we have $P(Yx,-)=[C^{op},Set](Yx,-)=ev_{x}$ the evaluation functor at x which preserves all colimits, so that left Kan extensions along Yoneda preserve all colimits.

Or if each $P(Yx,-)$ preserves filtered colimits then left Kan extensions along Y preserve filtered colimits.

I think this is all well known but don't know a reference.

Answer 2

$F$ preserving colimits doesn't imply that $\text{Lan}_Y(F)$ preserves colimits, even if all the categories are cocomplete.

Consider, for example, the case $C = D$ and $F = 1_C$. Then the left Kan extension $\text{Lan}_Y(1_C)$ exists if and only if $Y$ has a right adjoint, and if it does exist, it is the right adjoint of $Y$. (This is Theorem X.7.2 of Categories for the Working Mathematician.) Of course, $1_C$ preserves colimits, but right adjoints usually don't.

(From your notation, I guess you're generalizing from the case where $P$ is the category of Presheaves on $C$ and $Y$ is the Yoneda embedding. In that case, as I bet you know, $\text{Lan}_Y(F) = - \otimes F$ not only preserves colimits but has a right adjoint.)

Answer 3

Let $(a_i: A_i\to A)_{i\in I}$ with $I\in Cat$ a universal cocone in a category $\mathcal{A}$, and let $H: \mathcal{B}\to \mathcal{A}$.

We ask when:

for any $F: \mathcal{B}\to \mathcal{C}$ such that:

exist $L:=Lan_H F$ punctually (or at least it exist for the objects $A,\ A_i$ of the above diagram i.e. exists $L(A_i):=\varinjlim_{(B, b)\in H\downarrow A_i} F(B)$ and $L(A):=\varinjlim_{(B, b)\in H\downarrow A} F(B)$).

we have that $L(A)=\varinjlim_i L(A_i)$.

Consider the colimit category $\widehat{HA}:=\varinjlim_i H\downarrow A_i$, the functors $F\circ \pi_{A_i}:H\downarrow A_i\to \mathcal{B}\to \mathcal{C}$ induce a funtor $\hat{F}: \varinjlim_i H\downarrow A_i \to \mathcal{B}$ and is not hard verify that $\varinjlim_i L(A_i)=\varinjlim_i \varinjlim_{(B, b)\in H\downarrow A_i} F(B)= \varinjlim_{(B, b)\in \widehat{HA}} F(B)= \varinjlim \widehat{F}$.

Then the natural morphisms $\phi: \varinjlim_i L(A_i)\to L(A)$ is induced by the natural functor $\Phi: \widehat{HA}\to H\downarrow A $, then $\phi$ is a isomorphism (for any $F$ such that..) iff the functor $\Phi$ is final i.e. iff each morphism $H(B)\to A $ has a factorization on some $H(B')\to A_i\to A$ (through a morphism $B\to B'$) and two such factorization are connected in $H\downarrow A$.