Subobjects correspond to subgroups / subrings / submodules

Question

I read in MacLane that subobjects do correspond to subgroups and etc in standard categories. I don't understand why. So, a subobject of $a$ is a class of monic arrows $u : s \rightarrow a$ such that for any other arrow $v : s^\prime \rightarrow a$ from the class there's an isomorphism $\theta : s \rightarrow s^\prime$ such that $u = v \theta$. Now let's say we're in the category of abelian groups. There should be a 1-1 correspondence between subobjects of $a$ and subgroups of $a$. Let's say I have a monic $u : s \rightarrow a$. I relate to it the subgroup $\text{Im} \, u$, so I have an arrow $\imath : \text{Im} \, u \rightarrow a$. It's clear that I can find $v : S \rightarrow \text{Im} \, u$ such that $u = \imath v$. However, monics in the category of abelian groups may not be injective, so this $v$ may not be invertible (in the notation of MacLane, I can only show that $\imath \leq u$). How can I show that $\imath \equiv u$? I.e. how can I show that there is a map $v^\prime : \text{Im}\, u \rightarrow S$ such that $uv^\prime = \imath$?

Answer 1

First of all: Monics in the category of abelian groups are exactly the injective homomorphisms of groups.

"I read in MacLane that subobjects do correspond to subgroups and etc in standard categories. I don't understand why.":

If you define subobjects of $X \in \mathcal{C}$ as monomorphisms with codomain $X$, then this notion would not reflect the classical notion of subsets, subgroups etc. That is because you get too many different (categorical) subobjects that correspond to the same "substructure". Consider for example the singleton set $X = \lbrace x \rbrace$. The set $X$ has two subsets, namely the empty set $\emptyset$ and itself. We can easily find more monics though as for example $\emptyset \rightarrow \lbrace x \rbrace$, $\lbrace 1 \rbrace \rightarrow \lbrace x \rbrace$, $\lbrace 2 \rbrace \rightarrow \lbrace x \rbrace$ and $\lbrace 3 \rbrace \rightarrow \lbrace x \rbrace$, $...$ are monics (= injective maps in Set). Therefore we want to see all the latter ones (actually all the maps from singletons to $X$) as the same, namely as $\lbrace x \rbrace \rightarrow \lbrace x \rbrace$. That is the reason for taking the isomorphism classes of monics.

Try to understand the case of abelian groups once more and I will help you in case of problems.