# If an abelian category has a generator then it is well-powered

The title of my question is Proposition 3.35 in Freyd's Abelian categories. The proof says

If $G$ is a generator and $A$ is any object, then a subobject $A' \longrightarrow A$ is distinguished by the subset $(G,A') \subseteq (G, A)$.

What does "distinguished by" mean and why does this prove the proposition?

By "generator" Freyd surely means "strong generator". Anyway, let $A' \to A$ be a monomorphism, and consider the map $\textrm{Hom}(G, A') \to \textrm{Hom}(G, A)$. This is certainly injective, by the definition of monomorphism, so we may identify $\textrm{Hom}(G, A')$ with a subset of $\textrm{Hom}(G, A)$.

Suppose $A'' \to A'$ is another monomorphism. Then we get a chain of subsets $$\textrm{Hom}(G, A'') \subseteq \textrm{Hom}(G, A') \subseteq \textrm{Hom}(G, A)$$ and the definition of strong generator implies that $\textrm{Hom}(G, A'') = \textrm{Hom}(G, A')$ if and only if $A'' \to A$ and $A' \to A$ are isomorphic as subobjects of $A$. But what if $A' \to A$ and $A'' \to A$ are not comparable? Well, then take their pullback (which exists in any abelian category) and then use that as a bridge to compare $A' \to A$ and $A'' \to A$. The conclusion is, $\textrm{Hom}(G, A') = \textrm{Hom}(G, A'')$ as subsets of $\textrm{Hom}(G, A)$ if and only if $A' \to A$ and $A'' \to A$ are isomorphic as subobjects of $A$.

In particular, since $\textrm{Hom}(G, A)$ only has a set of subsets, $A$ only has a set of (isomorphism classes of) subobjects.

• Is it true that a Grothendieck category always has a strong generator? I'm just asking because I am sure I read somewhere that a Grothendieck category is always well-powered Jan 14 '13 at 17:17
• One definition of Grothendieck category asks that there be a generator. Since Grothendieck categories are locally small and cocomplete, by the remarks here, any generator is automatically a strong generator. Jan 14 '13 at 18:00

I'm not sure if the answer below is more well-suited to be a comment or as an answer, but I think that it might be helpful to include a short proof that in the setting of Freyd's book that the assertion can be proved directly without assuming that $$G$$ is strong. (but may only work in abelian categories)

Case 1 : Comparable Objects

By the definition of a generator (not necessarily strong), we see that the functor $$\operatorname{Hom}(G,-)$$ is left exact and faithful. For clarity, let us call this functor $$F$$.

Suppose we are given a proper inclusion $$A'\rightarrow A$$, we may extend the inclusion to an exact sequence $$0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$$ and apply the functor $$F$$.

If we were to have an exact sequence $$0\rightarrow FA'\rightarrow FA\rightarrow 0$$ then one concludes by faithfulness of $$F$$ that $$A\rightarrow A''$$ is just $$0$$ and get a contradiction. This shows that $$FA'$$ is a proper subset of $$FA$$ whenever $$A'$$ is a proper subset of $$A$$.

Case 2 : Incomparable Objects

This is done in Zhen Lin's answer.

(To be more precise : one can show that $$F$$ "commutes with" $$\cap$$ in some sense)