A problem on the union of subobjects

Question

I'm sorry for the silly doubt.

I have the following problem. Let $\mathfrak{C}$ be a category with $(Epi,Mono)$-factorizations, i.e. every morphism $f$ admits a unique $(Epi,Mono)$-factorization $f=\iota \circ p$.

Let $[\iota_i]$ (for $i \in I$) be an arbitrary collection of subobjects of some object $C$ and let $[\iota]$ be their union (suppose it exists).

Let moreover $f: C \to C'$ a given morphism and assume that $f \circ \iota=q \circ p$ is the $(Epi,Mono)$-factorization of $f \circ \iota$. Knowing that the $(Epi,Mono)$-factorizations of $f \circ \iota_i$ are $f \circ \iota_i=q_i \circ p_i$ for every $i \in I$, is it true that the subobject $[q]$ is the union of the subobjects $[q_i]$ of $C'$?

Answer 1

Yes it is true. Since the proof is a bit technical and could use a few diagrams (which are hard to write down here) I only sketch it to you and you can fill in the details as a very nice exercise.

1. First think about the universal property of the union of subobjects. It should be the smallest subobject $L$ containing all the given subobjects $Q$ in the sense that it is contained in every subobject $W'$ of $C'$, which itself contains all of the given subobjects $Q_i$.

2. Thus consider an arbitrary subobject $W'$ of $C'$, which contains all of your subobjects $Q_i$.

3. Pull it back along $f$ to get a subobject $W$ of $C$ (pullbacks of monos are monos).

4. Use the universal property of the pullback to note that $W$ contains all of your subobjects $J_i$ of $C$ (you also have to use the fact, that if a composite is monic, the first arrow is monic).

5. Use the universal property of the union of the $J_i$ to note that $W$ contains $J$.

Now we have to relate all of this to $W'$ and $C'$.

6. Consider the image factorization $T$ of $W \rightarrow W'$ and note that composing it with $W'\rightarrow C'$ gives the image factorization of $W \rightarrow C \rightarrow C'$. You can think of $T$ as the image of the preimage of $W$.

7. Now consider the image factorization $Q'$ of the composite $J \rightarrow W \rightarrow T$. Now note that composing it with the inclusion of $T$ into $C'$ gives an epi-mono factorization of the composite $J \rightarrow C \overset{f}\rightarrow C'$. But by definition $Q$ is this epi-mono factorization, so we obtain an isomorphism $Q \cong Q'$.

8. We conclude $Q$ is a subobject of $W'$ as desired.