# Abelian categories with generator objects are locally small

In the book "Rings of Quotients" by Bo Stenström, Proposition 6.6 on page 94 says: "If $$\mathbf{C}$$ is an abelian category containing a generator $$U$$, then $$\mathbf{C}$$ is locally small". The proof is made by establishing an injective correspondence between the class of sub-objects of a given object $$C$$ and the subsets of the set $$\operatorname{Hom}(U,C)$$.

Here is a sketch of the proof: "Given a sub-object $$\beta:B\to C$$, let $$\langle\beta\rangle$$ be the set of morphisms $$U\to C$$ which can be factored through $$\beta$$. If $$\beta':B'\to C$$ represents another sub-object we must show $$\langle\beta\rangle\neq\langle\beta'\rangle$$. Since $$\beta$$ and $$\beta'$$ are not equivalent, it can not occur that both monomorphisms $$\gamma: B\cap B'→B$$ and $$\gamma':B\cap B'→B'$$ are isomorphisms; say $$\gamma$$ isn't, so $$\operatorname{coker}\gamma\neq0$$. Hence there exists $$\alpha:U\to B$$ such that $$(\operatorname{coker}\gamma)\alpha\neq0$$"...- and the proof finishes by saying: "Then $$\alpha$$ can not be factored over $$B'$$, so $$\langle\beta\rangle\neq\langle\beta'\rangle$$."

I don't understand this final sentence. Why can $$\alpha$$ not be factored over $$B'$$, and why should this imply $$\langle\beta\rangle\neq\langle\beta'\rangle$$? Thanks.

• Don't you mean "well-powered" rather than locally small ? It seems like you're using that $Hom(U,C)$ is a set, implicitly assuming $\mathbf C$ is locally small May 5 '20 at 14:49
• No, I mean locally small, which mean that given any object C, the class of sub-objects B→C is a set. While by definition, given any two objects U and C in any category, the class of morphisms U→C is always a set. May 5 '20 at 15:35
• Ok well those aren't the standard definition. What you defined is usually called "well-powered" (see e.g. here : ncatlab.org/nlab/show/well-powered+category ) whereas locally small means that the hom-sets are small sets (see e.g. here : ncatlab.org/nlab/show/locally+small+category ). I guess it must be one of the author's idiosyncrasies May 5 '20 at 15:51
• Ok, in standar definitions the abelian category is locally small and I want to see, that is well powered provided contains a generator U, i.,e an object such that representable functor Hom(U,_) is faithful, I hope this be agree with standar definitions... May 5 '20 at 16:24
• Using "locally small" for what is now usually called "well-powered" wasn't particularly idiosyncratic for the time. It was once quite common terminology and used in some well-known early books such as Mitchell's "Theory of Categories". In Mac Lane's "Categories for the Working Mathematician" he explicitly states that he avoids using the term "locally small" at all because of the two conflicting meanings. May 6 '20 at 14:44

Since $$\alpha$$ is a map $$U\to B$$, and there are no maps $$U\to B'$$ or $$B'\to B$$ mentioned, it's not clear how $$\alpha$$ could factor through $$B'$$.
Almost certainly, Stenström means that the composition $$\beta\alpha: U\to C$$ does not factor through $$\beta':B'\to C$$.