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Let $X$ denote an object of a category.

Usually, we define that the subobjects of $X$ are equivalence classes of monomorphisms targeting $X$ such that for all $m,m' : M,M' \rightarrow X$ we have that $[m]=[m']$ iff there is an isomorphism $i : M \rightarrow M'$ making the obvious diagram commute.

Alternatively, we could define that the subobjects of $X$ are equivalence classes of morphisms targeting $X$ (not necessarily monic) such that for all $f,f' : F,F' \rightarrow X$ we have that $[f]=[f']$ iff there is a exist morphisms $i,i' : M \rightarrow M'$ (not necessarily inverses of each other) such that the two resultant diagrams commute.

Is there some sense in which these two definitions are "the same"?

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It's not an "alternative" if you can't show it to be equivalent. Consider the category of abelian groups, and take $X = \mathbb{Z} / 2 \mathbb{Z}$; $\mathbb{Z} \to \mathbb{Z} / 2 \mathbb{Z}$ is not (strongly) connected to any subobject of $\mathbb{Z} / 2 \mathbb{Z}$. – Zhen Lin Sep 16 '13 at 16:16
up vote 4 down vote accepted

As Zhen Lin points out in the comments, not every subobject in the second sense is a genuine subobject. For those whom are interested, this article refers them as "weak subobjects."

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