Supremum of a family of subobjects in an abelian category

Question

So this question follows on from an earlier one here. Suppose we are in a cocomplete abelian category. Say we have a (small) family of subobjects of some object $A$, which we will write as $\{ \mu_{i} : A_{i} \hookrightarrow A \}_{i \in I}$. Define the categorical sum as the image of the canonical morphism $$ \phi: \bigoplus_{i \in I} A_{i} \longrightarrow A. $$ Order the subobjects by inclusion: That is $A_{i} \leq A_{j}$ if there is a monomorphism $u: A_{i} \rightarrow A_{j}$ compatible with the inclusions $\mu_{i}$ and $\mu_{j}$). Then it is not hard to find that $\text{im}(\phi)$ is a subobject of $A$ and provides an upper bound for this family under the partial ordering. However, it seems to be assumed in the literature that this is in fact a supremum. Is anyone able to help me show this? In other words, if $B$ is some other subobject of $A$ acting as an upper bound, I need to find a morphism, $$ \rho: \text{im} (\phi) \longrightarrow B $$ compatible with the subobject structure. I feel like this should follow from universal properties of the image and direct sum, but I haven't been able to get it. I also tried to use epi-mono factorization properties. Is it actually true if the family isn't directed? Any help is appreciated.

Answer 1

Since $B$ is an upper bound for each $A_i$, they all factor through $B$: there are maps $\nu_i:A_i\to B$ such that the composition $A_i\to B\to A$ is the inclusion $A_i\to A$. These maps $\nu_i$ combine to give a map $\nu:\bigoplus A_i\to B$ whose composition with $B\to A$ is $\phi$. Thus the image of $\phi$ factors through $B$.

Answer 2

It is true !

Assume $\iota : B\hookrightarrow A$ is a subobject and for each $i\in I$ we have an inclusion $\delta_i : A_i\hookrightarrow B$ such that the obvious diagram commutes ($\iota\circ \delta_i = \mu_i$)

Then we also get an induced morphism $\psi: \displaystyle\bigoplus_{i\in I} A_i \to B$. But of course, since the morphisms were coherent in the beginning, the uniqueness part of the definition of coproduct makes it clear that $\iota\circ \psi = \phi$. Indeed, for each $i\in I$, $\iota\circ \psi\circ m_i = \iota\circ \delta_i= \mu_i = \phi\circ m_i$, where $m_i : A_i \to \displaystyle\bigoplus_{j\in I} A_j$ is the inclusion.

$\iota \circ \psi = \phi$, with $\iota$ monic. So if we write $\psi$ in its epi-mono-factorisation, we get an epi-mono factorisation for $\phi$. These are essetially unique and so $\text{im}(\phi) \hookrightarrow A$ factors through this, in particular it factors through $B$.