Natural transformation is a mono iff the components are.

Question

Could someone please give an honest proof of the fact that a natural transformation in the functor category $[\mathcal{C},\mathbf{Set}]$ is a monomorpism if and only if each of the components are monomorphisms?

I have seen it in many places, but never seen any proof and embarrasingly I can not do it myself.

Answer 1

This is just a case of spelling out a couple of definitions. What does it mean for a natural transformation in $[C, Set]$ to be a monomorphism?

Let $F:C \rightarrow Set$ and $G: C\rightarrow Set$ be functors and let $\eta : F\Rightarrow G$ be a natural transformation between them. Now, $\eta$ is a monomorphism if for any functor $H: C\rightarrow Set$ and any two natural transformations $\gamma : H \Rightarrow G$ and $\gamma' : H\Rightarrow G$,

$$\eta \circ \gamma = \eta \circ \gamma' \Rightarrow \gamma = \gamma'$$

For two natural transformations to be equal, they must be equal at each component, so the above statement means that:

For all $c\in Set$ ,

$$(\eta \circ \gamma)_c = (\eta \circ \gamma')_c \Rightarrow \gamma_c = \gamma'_c$$

Now (vertical) composition of natural transformations is given component-wise, hence, for all $c\in C$

$$\eta_c \circ \gamma_c = \eta_c \circ \gamma'_c \Rightarrow \gamma_c = \gamma'_c$$

Which is precisely the statement that each component of the natural transformation is a monomorphism.

Answer 2

It's easy to see via a direct application of the definition of monos that if $\tau \colon F \to G$ is a natural transformation between two functors $F,G \colon \mathbf C \to \mathbf D$ such that $\tau_c \colon F(c) \to G(c)$ is a monomorphism for every $c \in \mathbf C$ then $\tau$ is a monomorphism between the functors $F$ and $G$. Indeed if that's the case then for every pair $\sigma,\sigma' \colon H \to F$ of natural transformations if $$\tau \circ \sigma = \tau \circ \sigma'$$ then for every $c \in \mathbf C$ we have $$\tau_c \circ \sigma_c = \tau_c \circ \sigma'_c$$ and by mono-property of $\tau_c$ it follows that $\sigma_c=\sigma'_c$, and so $\sigma=\sigma'$.

Edit: Applying this to the case where $\mathbf D$ is the category $\mathbf {Set}$ we obtain that if $\tau \colon F \to G$ is a natural transformation between two $\mathbf{Set}$-valued functors ($F,G \colon \mathbf C \to \mathbf {Set}$) such that $\tau_c$ is a mono for every $c \in \mathbf C$ then $\tau$ is a mono.

We are going to prove the other implication through an application of the yoneda lemma, which establishes the existance of an isomorphism $$[\mathbf{C}(X,-),F]\stackrel{y}{\cong} F(X)$$ which is natural both in $X$ and (most important for our purpose) $F$.

From the naturality of yoneda bijection, in the functor part, we have for every $c \in \mathbf C$ the commutativity of the following diagram of sets $$\require{AMScd} \begin{CD} [\mathbf{C}(c,-),F] @>y>> F(c)\\ @V{[\mathbf C(c,-),\tau]}VV @V{\tau_c}VV \\ [\mathbf{C}(c,-),G] @>y>> G(c)\\ \end{CD}$$ where $[\mathbf C(c,-),\tau]$ is the image of $\tau$ via the $\hom$-functor $[\mathbf C(c,-),-] \colon \mathbf {Func}(\mathbf C,\mathbf{Set}) \to \mathbf {Set}$.

$\hom$-functors preserve monos so if $\tau$ is mono then also $[\mathbf C(c,-),\tau]$ is a mono, and so $\tau_c = y \circ [\mathbf C(c,-),\tau] \circ y^{-1}$ must be a mono too, since it's obtained via composition of three monos (remember that $y$ and $y^{-1}$ are isomorphisms and isomorphisms are monos).

That should complete the proof.

Answer 3

The complete Q&A thread has been linked by Zhen Lin, but I will reproduce the answer here for your convenience.

1- if all components of a natural transformation are mono (resp. epi), then the natural transformation is mono (resp. epi)

2- if a natural transformation $\tau$ between functors $F,G :\mathcal{C} \to \mathcal{D}$ is mono (resp. epi) and $\mathcal{C}$ is small and $\mathcal{D}$ has pulbacks (resp. pushouts), then each component of $\tau$ is is mono (resp epi).

Statement 1 has been proved by Mossa and Revell in their answers. Statement 2' s proof can be found in section 2.15 of the Handbook of categorical algebra, volume I (by F. Borceux) pages 87--90. In particular, their Corollary 2.15.3 covers the case for small $\mathcal{C}$

Since $Set$ has pullbacks and pushouts, you can apply this theorem to $\tau$ in [$\mathcal{C}$,$Set$]