# When colimit of subobjects is still a subobject?

What are the conditions on a category (or on a certain object) that will guarantee that the colimit of a family of subobjects of a given object is a subobject of the same object?

Update: To clarify the question - let $C$ be a category with arbitrary colimits. Consider a family $\mathcal{I}=\{X_i\to X\}$ of subobjects of $X$, such that $\mathcal{I}$ is a semilattice w.r.t inclusion relation of subobjects. Then one can take the colimit $lim_{\to \mathcal{I}}(X_i)$ in $C$. What are the conditions on the category $C$ that guarantee that the canonical map $lim_{\to \mathcal{I}}(X_i)\to X$ is a monomorphism, i.e. $lim_{\to \mathcal{I}}(X_i)$ is a subobject of X?

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What do you mean by the colimit of a family of subobjects? In what category, the ambient category or the category of subobjects? If the former, in what sense is being a subobject a property rather than a structure? If the latter, the colimit, if it exists, is by definition a subobject, so are you asking when colimits exist in subobject categories? –  Qiaochu Yuan Aug 20 '13 at 6:39
@Qiaochu: it is the former. I edited the question and, hopefully, made it clear. –  user35953 Aug 20 '13 at 12:26
What kind of semilattice: join or meet? –  Zhen Lin Aug 20 '13 at 12:45
@ZhenLin: meet. For join, I guess, one would need some additional assumptions. Although, I believe, you should know this topic better~ –  user35953 Aug 20 '13 at 13:11
If you said join then the diagram would have been filtered, and then what you want is often true. For meet, things are harder, but I think what you want is true in, say, a Grothendieck topos. –  Zhen Lin Aug 20 '13 at 16:38

I assume that your question is: If $\{X_i \to X\}$ is a diagram of subobjects of $X$, and $\mathrm{colim}_i X_i$ is a colimit in the ambient category, when is the induced morphism $\mathrm{colim}_i X_i \to X$ again a monomorphism and therefore exhibits the colimit as a subobject of $X$?
But many categories enjoy the following property: A directed colimit of monomorphisms is a monomorphism. Notice that for abelian categories this is part of Grothendieck's axiom AB5. If this property is satisfiesd, and the diagram $\{X_i\}$ is directed, then of course $\mathrm{colim}_i X_i \to \mathrm{colim}_i X = X$ is a monomorphism. For example, you can consider directed colimits of subrings of a ring, of subfields of a field, of subspaces of a space, etc.