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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
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@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

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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 $X$?

Well without restrictions, of course this fails terribly. Consider any of your favorite concrete algebraic or topological categories and look at discrete colimits, i.e. coproducts. It also fails for coequalizers.

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.

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yes, your assumption is correct. But I only need to consider colimits w.r.t. inclusion relation, thus only partially ordered, not directed. Also I don't want to restrict to abelian categories. So my question is about minimal possible conditions on the ambient category. –  user35953 Aug 20 '13 at 12:31
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a) You misunderstood the assumption of being directed. It refers to the index category of the diagram. b) I don't restrict to abelian categories. –  Martin Brandenburg Aug 20 '13 at 16:14

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