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Consider the category $mon(C)$ (objects as monics in $C$) as full subcategory of the $Arrow(C)$ . We know that $Arrow(C)$ is finitely complete and cocomplete, assuming that $C$ has limits as well as colimits. Does $mon(C)$ (finitely) cocomplete?

For example consider coproducts: For objects $m_1:A \rightarrow C$ and $m_2:B \rightarrow D$ in $Arrow(C)$, the coproduct is given by $[m_1, m_2]:A+B \rightarrow C+D$ (using the couniversal property of $A+B$). Does $m_1$, $m_2$ monic implies $[m_1,m_2]$ being monic?

I think it is not true in general. If it is assumed that $C$ is a topos, does that help in cocompleteness of $mon(C)$ in anyway? (For $C$ topos, $mon(C)$ will have exponents, but not subobject classifier).

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The full subcategory of monomorphisms is closed under limits inside the category of morphisms. (Easy exercise.)

In general, colimits in the category of monomorphisms are not the same as colimits in the category of morphisms (assuming they even exist). But there is sometimes a relation. Assuming the category in question has image factorisation:

  • The inclusion of the subcategory of monomorphisms has a left adjoint.
  • If the category has colimits, then so too does the category of monomorphisms.
  • To calculate a particular colimit in the category of monomorphisms, first take the colimit in the category of morphisms and then apply the left adjoint to it.

In particular, if your category is a topos, then you are in the above situation.

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  • $\begingroup$ Does the Easy exercise follow from the fact that monos can be characterized as a limit property and limits commute with limits? $\endgroup$ – Rachmaninoff Nov 20 '14 at 1:18
  • $\begingroup$ That's one way of seeing it, yes. $\endgroup$ – Zhen Lin Nov 20 '14 at 8:45
  • $\begingroup$ @ZhenLin Do you have any references to know more about such categories? $\endgroup$ – Anuj More Dec 6 '14 at 17:22
  • $\begingroup$ Not really. Just read category theory in general, I suppose. $\endgroup$ – Zhen Lin Dec 6 '14 at 18:09

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