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?

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.
