Question on Category Theory injective morphism

Question

I have a basic understanding in Category Theory but haven't had any exposure to Modules but this is a question from last year's paper.

Show that a morphism $u:M \to L$ in a category $\mathscr C$ of modules over a ring R (or the category of sets) is injective iff for any $X$ of $\mathscr C$ and $v \in Hom_R(X,M)$ the map $Hom_R(X,M) \to Hom_R(X,L), v\mapsto uv$ is injective.

Also is it the case that map $u$ is surjective iff $v \mapsto uv$ is surjective? If so, how do I prove this?

Thanks in advance for any replies.

Answer 1

From the elementary level of the question it is clear that no reference to the RAPL (right adjoints preserve limits) theorem is required (nor welcome). It is also clear that the question is not really specific to modules, since it refers also to the category of sets.

The author of the question simply wants you to be able to translate a elementary algebra definition (injection) which is made in terms of elements of a set (or module or group, or ring) to a categorical one (monomorphisms) which is made in terms of morphisms.

So , how do you translate $x \in X$ into category language which only uses objects and morphisms? Very simple: in the category $\mathcal{Set}$ every element $x$ of a set $X$ is in bijenctive correspondence with a morphism (that is: a function) $x^\prime$ from a simgleton to $X$. This morphism takes the unique element in the singleton to $x \in X$.

In the category of $Mod_R$, instead of singleton you use the module $R$ and the morphism (that is: a linear transformation) from $R$ to $X$ which takes $1$ to $x$ in module $X$.

In the category $Grp$ you would use the $Z$ group of integers and the morphism from $1$ to $x$ in group $X$

And so on , as long as you work in a category where an element of an object can be described as a special morphsm from a special object in the category.

Now whenever you want to translate a formula $f(x)$ into category language, you just write $f \circ x^\prime$.

Wth this simp,e translation rule, you can easily express injectivity in terms of morphisms and see that (in these categories) it is equivalent to monomorphism.

Answer 2

A map $u \colon M \to L$ (in any category $\mathscr C$) such that the induced map $\hom_{\mathscr C}(X,M) \to \hom_{\mathscr C}(X,L)$ is injective for all object $X$ in $\mathscr C$ is called a monomorphism.

You first question actually is :

Let $R$ be a ring. Show that the monomorphisms of the category of modules over the ring $R$ are exactly the injective maps of modules.

There is a really quick way to show it. First, it is clear that the injective maps of modules are monomorphisms. Conversely, the forgetful functor $U \colon R{-}\mathsf{Mod} \to \mathsf{Set}$ admits a left adjoint (namely, the free $R$-module functor), hence commutes with limits, and so takes monomorphisms to monomorphisms. The monomorphisms of $\mathsf{Set}$ being the injections, it concludes.

As for your second question, Zhen Lin already responded in the comments. However, you are probably interested in the notion of epimorphism : a map $u \colon M \to L$ of $\mathscr C$ is called an epimorphism when the induced map $\hom_{\mathscr C}(L,X) \to \hom_{\mathscr C}(M,X)$ is injective for all object $X$ in $\mathscr C$.