Definition of subobject in category theory?

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A subobject of $X$ in category theory is defined as a certain equivalence class of monomorphisms to $X$.

What I don’t understand about this definition, is that this doesn’t actually define an object in the category (as far as I know). It seems at first sight to me to be a type error? We are looking for an object, not an equivalence class of arrows? How does this define an object?

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1 Answer

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The object is the domain of the monomorphism; this is why IMO the correct definition should just be "a monomorphism," period. Taking equivalence classes doesn't buy you much, really.

If you haven't, it's worth working through special cases of this definition. In $\text{Set}$ you get subsets, in $\text{Grp}$ you get subgroups, and so on and so forth.

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