202512311455

Tags : Category Theory

Pointwise Kan Extensions

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Definition

When $E$ is locally small, a right kan Extensions|kan extension is a pointwise kan extension if it is preserved bt all representable functors $E(e,-)$.

Theorem

A right Kan extension of $F$ along $K$ is pointwise if and only if it can be constructed by the following:

$${\text{Ran}}_{K}F=\lim \left(d/K\stackrel{{\Pi }_d^K}{\to }C\stackrel{F}{\to }E\right)$$

in which case, this limit exists.

If ${\text{Ran}}_{K}F$ is pointwise, then by Yoneda Lemma and the defining universal property of Kan Extensions

$$\begin{array}{rl}E(e,{\text{Ran}}_{K}F)& \cong {\text{Set}}^D(D(d,-),E(e,{\text{Ran}}_{K}F))\\ & \cong {\text{Set}}^C(D(d,K-),E(e,F-))\\ & \cong \text{Cone}(E,F{\Pi }_d^K)\end{array}$$

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References

• Kan Extensions
• Yoneda Lemma