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Stream: learning: questions

Topic: Proving that the collection of functors between two categ...

Julius Hamilton (Aug 30 2024 at 22:55):

There are a few small questions that have come up as I have tried to prove this.

For a natural transformation $\eta : F \Rightarrow G$ ($F, G: C \to D$), we have a family of morphisms $\{\eta_X\}$ in $D$, from the objects $F(X)$ to the objects $G(X)$ for all $X \in C$.

How do we know that a morphism always exists in $D$ from $F(X)$ to $G(X)$?

I’m going to guess that it follows from $F$ and $G$ being functors.

Julius Hamilton (Aug 30 2024 at 22:59):

It seems like the definitions of natural transformation I have seen do not choose to express it as a collection of morphisms from the image of $F$ to the image of $G$. Is there a reason people would avoid expressing it this way?

Joshua Meyers (Aug 30 2024 at 23:00):

Julius Hamilton said:

How do we know that a morphism always exists in D from F(X) to G(X)?

We don't. The claim that $\eta: F\Rightarrow G$ is a natural transformation implies the claim that for any object $X$ of $C$, a morphism exists in $D$ from $F(X)$ to $G(X)$.

Joshua Meyers (Aug 30 2024 at 23:02):

Julius Hamilton said:

It seems like the definitions of natural transformation I have seen do not choose to express it as a collection of morphisms from the image of $F$ to the image of $G$. Is there a reason people would avoid expressing it this way?

Suppose $F(X)=F(Y)$, and $G(X)=G(Y)$. Then we would still need two morphisms $\eta_X$ and $\eta_Y$, and they might even have to be distinct in order for the naturality condition to be true. But if we thought in terms of the images of $F$ and $G$, we wouldn't notice that these points have to be counted twice because they just occur once in the images.

Julius Hamilton (Aug 30 2024 at 23:03):

Ok, those are really good answers, thank you.

Julius Hamilton (Aug 30 2024 at 23:22):

When we prove that the collection of functors $[C, D]$ forms a category, with natural transformations as morphisms, we don’t know the morphism structure of the functor category in general. Maybe it’s a discrete functor category, or an indiscrete one. But we should define a composition operation for natural transformations. A natural transformation $\eta$ between functors $F$ and $G$ is a family of morphisms $\{\eta_X\}$ in the target category ($D$), $X \in C$, such that $\eta_Y \circ F(f) = G(f) \circ \eta_X$ for all $(f : X \to Y) \in C$.

Is it conceivable that we might have multiple choices for how to define the composition operation, as long as it is associative and has a left-right-identity?

John Baez (Aug 31 2024 at 00:02):

Julius Hamilton said:

How do we know that a morphism always exists in $D$ from $F(X)$ to $G(X)$?

We don't. Make up an example where there's not.

(Hint: you can let $C$ have one object and one morphism, and let $D$ have two objects and two morphisms.)

Of course in this case there cannot exist a natural transformation from $F$ to $G$.

Julius Hamilton (Aug 31 2024 at 04:25):

nattrans.png

Here it is

Julius Hamilton (Aug 31 2024 at 04:27):

For all $X$, there must exist a component $\eta_X: F(X) \to G(X)$. But no such morphism exists; therefore, there are no natural transformations $\eta : F \Rightarrow G$.

Julius Hamilton (Aug 31 2024 at 04:46):

I am working through a proof that the functors $[C, D]$ form a category, with David Egolf.

functorcategories.jpg

I am curious, if there is a deeper reason why this is so? It’s interesting that the structure of a category re-appears so readily. It reminds me of what John Baez said about exponential objects, where we want say, the collection of homomorphisms from a group to themselves form a group.