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The idea of a replete subcategory to me seems very analogous to the idea of a characteristic subgroup, as both are, in some sense, subobjects invariant under a notion of equivalence (categorical equivalence for replete subcategories, automorphisms for characteristic subgroups). This led me to the question: Is there a notion in category theory that is similarly analogous to that of a normal subgroup in group theory?

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    The sense in which replete subcategories are "invariant" under equivalences is quite different from the sense that normal subgroups are "invariant" under automorphisms. So it is not at all clear what analogy you are going for here. – 2017-02-03
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    You might think of normal subgroups as being those subgroups that you can quotient by and still get a group. That is, the equivalence relation defined by the subgroup is good enough to give you a group when you quotient by it. You can take quotients by equivalence relations categorically. So you might ask under what conditions a subcategory gives "a good enough equivalence relation" to give a quotient category... – 2017-02-03
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    Yes, I know the analogy isn't perfect (although, I was hoping that possibly there was a deeper connection between the concepts that one could express formally), but what John Martin is saying is essentially what I am looking for. – 2017-02-03
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    There *is* a notion of a normal subobject in a category, but I don't know of any example in the case of categories and I suspect it is useless because important properties of categories usually refer to natural transformations at some point. – 2017-02-03
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    @StefanPerko Really, of a normal subobject in an arbitrary category? – 2017-02-03
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    @KevinCarlson Sorry. I forgot to mention: You probably want to have finite limits (that's not much though and I'm not sure whether it is *really* necessary, but it is certainly convenient), see "Mal'cev, Protomodular, Homological and Semi-Abelian Categories" by Borceux and Bourn. - Although, I can't say anything about the usefulness of this concept outside of at least protomodular categories. - But it is nice, that normal $\Rightarrow$ mono (even in general categories with finite limits), even though it "mono" not obviously part of the definition of "normal". – 2017-02-03

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