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functions; but not all injective continuous functions are subspace embeddings. In the category of rings, the inclusion is an epimorphism but is not the quotient of by a two-sided ideal. To get maps which truly behave like subobject embeddings or quotients, rather than as arbitrary injective functions or maps with dense image, one must restrict to monomorphisms and epimorphisms satisfying additional hypotheses. Therefore, one might define a "subobject" to be an equivalence class of so-called "regular monomorphisms" (monomorphisms which can be

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