I am trying to prove a collection of results about subobjects of abelian categories. With a little work, it’s not hard to show that the subobjects form a distributive lattice. Note: There is a concrete question I would like answered at the end of this post.
Given an object $M$ and two monomorphisms $a: A \rightarrowtail M$ and $b: B \rightarrowtail M$, we may construct the meet $a \cap b$ as $ \text{ker}(\text{cok}(b) \circ a)$ and the join $a \cup b$ as $\text{ker}(\text{cok}(a \oplus b))$, where $a \oplus b : A \oplus B \rightarrow M$ is the canonical map out of the biproduct. Denote the domains of these maps by $A \cap B$ and $A \cup B$ respectively.
These constructions are easily seen to be generalizations of the intersection and “internal sum” of submodules. I am interested in figuring out how similar/dissimilar these constructions are in general.
For example, it is clear that while infinite families of submodules always have an intersection, this is not true in all abelian categories.
I would like to prove some similarities though. For example, I would like help resolving the following question:
If we have submodules $A,B \subseteq M$ and a linear map $\phi : N \rightarrow M$, we may pullback $A$ and $B$ to submodules $\phi^{-1}(A)$ and $\phi^{-1}(B)$ of $N$. This can easily be accomplished in an arbitrary abelian category by taking pullbacks. However in the module case, it is clear that if $A \cup B \cong M$, then $\phi^{-1}(A) \cup \phi^{-1}(B) \cong N$. Does this extend to all Abelian categories as well?
I have tried multiple approaches, but nothing has panned out. Help and suggestions would be appreciated. This feels like it should be true, and I’m sure there’s some way to set up a commutative diagram and apply the 4 or 5 lemma, but I don’t see it. Thanks!