# intersections in abelian category

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?

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).

• Many thanks John!, this is very useful :) – Diego Feb 7 '13 at 2:42
• @Diego No problem! Freyd's book is a lovely little gem. I highly recommend it. – John Myers Feb 7 '13 at 2:43

Just for the general interest, I have found an interesting book for the framework in which one could define intersections:

Categorical structure of closure operator with application to topology, algebra and discrete mathematics, Dikranjan D.N., Tholen W.

Definition of intersection p. 14-15

Equivalent conditions for the existence of intersections: Theorem p.17

The whole first chapter is about subobjects (+ first § of 5.2).

One more comment: i sometimes got a little bit confused when reading (always happens to me), I had the impression Dikrankan was defining the same thing again but the cause is that I didn't pay sufficiently attention to where the arrows belong to. E.g. an intersection is a pullback, but of subobjects.

For $A,B\subset C$ the intersection $A\cap B$ is the kernel of the canonical map $A\oplus B\rightarrow C$. Kernels exist in an abelian category.