Franklin's Notes

Natural transformations between representable functors

Given any two covariant representable functors $\mathsf{C}(X,-):\mathsf{C}\to\mathsf{Set}$ and $\mathsf{C}(Y,-):\mathsf{C}\to\mathsf{Set}$ and a morphism $f:X\to Y$ between the objects representing them, we can find a natural transformation $\alpha^{(f)}:G\implies F$ whose components $\alpha_Z^{(f)}:\mathsf{C}(Y,Z)\to \mathsf{C}(X,Z)$ between hom-sets are defined by pre-composition by $f$. Since the functors $\mathsf{C}(X,-)$ and $\mathsf{C}(Y,-)$ send morphisms $h:X\to Z$ and $h:Y\to Z$ to the respective pre-composition maps in $\mathsf{Set}$, naturality of $\alpha$ follows from the associativity property of function composition.

Exercise 1. (Riehl, Exercise 2.1.i) For each of the three functors describe the corresponding natural transformations between the covariant functors $\mathsf{Cat}\to\mathsf{Set}$ represented by the categories $\mathbb{1}$ and $\mathbb{2}$.

Proof. The covariant functors $\mathsf{Cat}\to\mathsf{Set}$ represented by $\mathbb{1}$ and $\mathbb{2}$ are naturally isomorphic to the functors $\text{ob}:\mathsf{Cat}\to\mathsf{Set}$ and $\text{mor}:\mathsf{Cat}\to\mathsf{Set}$, sending a small category to its set of objects and set of morphisms, respectively. Using the construction described above, we have natural transformations with components described as follows:

Why is this? For any object $X\in\mathsf{C}$, there is a morphism $\text{getOb}_X\in\mathsf{Cat}(\mathbb 1, \mathsf{C})$ sending the unique object of $\mathbb 1$ to the object $\mathsf C$. Similarly, for any morphism $f$ of $\mathsf{C}$, there is a morphism $\text{getMor}_f\in\mathsf{Cat}(\mathbb 2, \mathsf{C})$ sending the unique non-endo morphism of $\mathbb 2$ to $f$. Because of this, there are natural isomorphisms and Now, given a morphism $f$ of $\mathsf{C}\in\mathsf{Cat}$, we have the following equalities: and given an object $X\in\mathsf{C}$, we have Here is a commutative diagram describing this situation:





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