Let $\mathcal C$ be a category. Recall that a morphism $f : X \to Y$ is epi if $$\circ f: \hom(Y,Z) \to \hom(X,Z)$$ is injective for each object $Z \in \mathcal C$. ($f$ is mono if $f\circ : \hom(Z,X) \to \hom(Z,Y)$ is injective.)
Let $\mathcal C,\mathcal D$ be categories. Then $\hom(\mathcal C,\mathcal D)$, the collectional of all functors $\mathcal C \to \mathcal D$, is naturally a category, where the morphisms are natural transformations: if $F,G: \mathcal C \to \mathcal D$ are functors, a natural transformation $\alpha: F \Rightarrow G$ assigns a morphism $\alpha(x) : F(x) \to G(x)$ in $\mathcal D$ for each object $x \in \mathcal C$, and if $f: x \to y$ is a morphism in $\mathcal C$, then $\alpha(y) \circ F(f) = G(f) \circ \alpha(x)$ as morphisms in $\mathcal D$.
Given a natural transformation, can I check whether it is epi (or mono) by checking pointwise? I.e.: is a natural transformation $\alpha$ epi (mono) iff $\alpha(x)$ is epi (mono) for each $x$?
If not, is there an implication in one direction between whether a natural transformation is epi and whether it is pointwise-epi?
A more general question, one that I never really learned, is what types of properties of a functor are "pointwise" in that they hold for the functor if they hold for the functor evaluated at each object. E.g.: is the (co)product of functors the pointwise (co)product?
Best Answer
Theo, the answer is basically "yes". It's a qualified "yes", but only very lightly qualified.
Precisely: if a natural transformation between functors $\mathcal{C} \to \mathcal{D}$ is pointwise epi then it's epi. The converse doesn't always hold, but it does if $\mathcal{D}$ has pushouts. Dually, pointwise mono implies mono, and conversely if $\mathcal{D}$ has pullbacks.
The context for this --- and an answer to your more general question --- is the slogan
You have, let's say, two functors $F, G: \mathcal{C} \to \mathcal{D}$, and you want to compute their product in the functor category $\mathcal{D}^\mathcal{C}$. Assuming that $\mathcal{D}$ has products, the product of $F$ and $G$ is computed in the simplest possible way, the 'pointwise' way: the value of the product $F \times G$ at an object $A \in \mathcal{C}$ is simply the product $F(A) \times G(A)$ in $\mathcal{D}$. The same goes for any other shape of limit or colimit.
For a statement of this, see for instance 5.1.5--5.1.8 of these notes. (It's probably in Categories for the Working Mathematician too.) See also sheet 9, question 1 at the page linked to. For the connection between monos and pullbacks (or epis and pushouts), see 4.1.31.
You do have to impose this condition that $\mathcal{D}$ has all (co)limits of the appropriate shape (pushouts in the case of your original question). Kelly came up with some example of an epi in $\mathcal{D}^\mathcal{C}$ that isn't pointwise epi; necessarily, his $\mathcal{D}$ doesn't have all pushouts.