Quirkiness with Kan Extensions

Quirkiness with Kan Extensions

Given two functors $F : C \to D$ and $K : C \to D$left kan extension of $F$ along $K$ is a functor Lan$_kF : D \to E$ along with a natural transformation $\eta : F \to \text{Lan}_KF K$ universal in that for any pair $(G: D \to E, \gamma: F \to GK)$ there is a unique natural transformation $\alpha : \text{Lan}_KF \to G$ such that $\gamma = \alpha_K \eta$ as in the diagram

Our counterexample is meant to define intuition about this commutativity “up to universal natural transformation.” A properly commutating diagram may not actually be the Kan extension.  In other words, if $F$ factors through $K$ by some $H$ then $(H,1_H)$ is not necessarily the same as $(\text{Lan}_KF, \eta).$

Spoiler Alert: This is exercise 1.1.3. in [1]. If you want to find out for yourself, stop reading.

Take $1 = C, E=Set$ and $D$ arbitrary. Let $d \in D$ be an arbitrary object and let $d : 1 \to D$ denote the functor constant at $d.$ We take $F = \ast$ e.g. $F$ is the functor constant at some singleton set $\ast.$ Then, we can (by abuse of notation) write $\ast : D \to Set$ again for the functor constant at $\ast.$ Indeed we have $\ast = \ast \circ d.$ But, $\ast : D \to Set$ is not the (left) Kan extension of $\ast$ along $d.$

Claim: $(\text{Lan}_d\ast,\eta)=(\text{Hom}_D(d,-),1_D)$

Why is this true?  Because elements $x \in Fd$ are in bijection with natural transformations $x : \ast \to F.d$ By Yoneda’s lemma, these elements are in natural bijection with natural transformations $\text{Hom}_D(d,-) \to F.$ This is exactly the universal property of left Kan extensions.

References

[1] Riehl, E. Categorical Homotopy Theory. Cambridge University Press, 2014.