Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow A$ to $v:C\rightarrow A$ is a morphism $ w: B\rightarrow C$ such that $v\circ w=u$. We can define the intersecion of $u$ and $v$ as a product of $u$ and $v$, that is, it is a monomorphism $w:K\rightarrow A$ with morphisms $\pi_1:K\rightarrow B$ and $\pi_2:K\rightarrow C$ such that for any other monomorphism $w':K'\rightarrow A$ with morphisms $p_1:K'\rightarrow B$ and $p_2:K'\rightarrow C$ there exists a unique map $p:K'\rightarrow K$ such that $\pi_i\circ p=p_i$.
Is this the correct definition of intersections of subobjects in an abelian category? and if it is, which are the general conditions that we have to impose on $\mathcal{A}$ to ensure that every pair of subobjects of any object have intersection?
Best Answer
How are you defining subobject? Do you declare a monomorphism $B \to A$ to be a subobject of $A$ (like Mitchell), or do you define subobjects of $A$ to be certain equivalence classes of monics with target $A$ (like Mac Lane or Freyd)?
In any case, you have the right idea.
If we take the "subobjects are equivalence classes" definition, then recall that the class of subobjects of a given object has a natural partial order. If $u: B \to A$ and $v: C \to A$ represent two subobjects of $A$, we declare $u \leq v$ iff $u$ factors through $v$. The intersection of two subobjects is then their greatest lower bound with respect to this order (as it should be!).
Now, given an arbitrary abelian category, the intersection of any pair of subobjects (of a fixed object) always exists. This is Theorem 2.13 in Freyd's book (ftp://ftp.sam.math.ethz.ch/EMIS/journals/TAC/reprints/articles/3/tr3.pdf), and it's not a hard proof (it's the third theorem he proves about abelian categories).