# Can epi/mono for natural transformations be checked pointwise?

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?

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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

(Co)limits are computed pointwise.

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.

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In all the preceding answers the category $D$ is required to have pushouts / pullbacks in order to epi / mono being equivalent to pointwise epi / mono in functor categories $D^C$.

The discussion in On a corollary in Mitchell's book draw my attention to another important class of categories that usually doesn't have pushouts or pullbacks and where epi / mono is also equivalent to pointwise epi / mono in $D^C$: That is when $D$ is exact.

A category is exact, if

• it has a zero object
• kernels and cokernels exist
• every monomorphism is a kernel and every epimorphism is a cokernel
• every morphism can be written as a composition of an epimorphism followed by a monomorphism.

As a reference see Barry Mitchell: "Theory of Categories", II.11 Functor Categories.

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I guess you just need some way of using limits to detect monos. Pullbacks give one (via kernel pairs) and every mono being a kernel gives another. – Ben Millwood May 5 '13 at 11:35

The accepted answer is good. If you would like another reference see section 2.15 of the Handbook of categorical algebra, volume I (by F. Borceux) pages 87--90. In particular, their Corollary 2.15.3 tells us the following:

Let $F,G: \mathcal{C} \rightarrow\mathcal{D}$ be two functors where $\mathcal{C}$ is a small category and $\mathcal{D}$ has pullbacks. Then a natural transformation $\alpha : F \rightarrow G$ is monic in $\mathcal{D}^{\mathcal{C}}$ if and only if for each object $C\in \mathcal{C}$, $\alpha_C : F(C)\rightarrow G(C)$ is monic in $\mathcal{D}$.

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This is only a partial answer. Regarding your first question (mono/epi iff pointwise mono/epi): At least for the case where the target category $\mathcal{D}$ is $\mathbf{Set}$, it is true that pointwise mono/epi implies mono/epi, see p. 91 of the 1998 edition of Mac Lane.

As for the second question, the answer is that pointwise limits implies limits in functor categories, by the limit with parameters'' theorem (Theorem V.3.1 of Mac Lane).

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After the edit, I've noticed that Tom Leinster also answered the question. So now you have the answer of an expert :). I have decided not to delete my answer, hoping that it may help in some way. – user2734 Mar 12 '10 at 12:18

The answere is "Yes" if the caegory has pullbaks, otherwise no in general: Let $C$ a category and $I$ a diagram, If $C$ ha limits (of some kind) then the some happen for $C^I$ because you can make limits pointwise. Then if $m: F\Rightarrow G$ is Mono (in $C^I$ ) the pullabck of $(m,m)$ give the cospan $F\leftarrow^{1_F}F\to^{1_F}F$ and if $C$ has pullback then you can see what above componentwise, then any pullback of $(m(i), m(i))$ give the cospan $F(i)\leftarrow^{1_{F(i)} }F(i)\to^{1_{F(i)}}F(i)$ then any $m(i)$ is Mono. Now, let $C$ the follow category where no identity arrow are as: $f,g: A\to B$, $h: B\to C$ where $h\circ f= h\circ g$ then in $C^\to$ we have the Monomorphism $(f, h): g\to h$ but $h$ isnt Mono, by the way you get the not pointwise pullback of $(f,h), (f,h))$ give the cospan $g\to^1 g\\leftarrow^{1 } g$ (I take this argument from G Kelly: http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html).

Sorry for my poor English, and Latex , but I use xymatrix for make diagram and seems dont work here..

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+1 for the counterexample! – Hew Wolff Aug 13 '15 at 3:24