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As we know, we have monoid objects (or ring objects) in monoidal categories. For example, the monoid objects in $\mathrm{Vect}\; k$ are $k-$algebras.

I want to question that can we define a "subring object" to categorification the notion of "subalgebra"?

It seems hard to have the notion "sub" in categories. And I searched the term "relative category", but it is a little hard to understand and seems to have no relation to us.

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  • $\begingroup$ The most common notion of subobject in algebra is isomorphism classes of monomorphisms. Have you tried looking at subobjects in the category of monoids on $k\mathrm{Vect}$, for example? $\endgroup$
    – Naïm Favier
    Commented Nov 13 at 8:38
  • $\begingroup$ That's an answer, not a comment. $\endgroup$
    – Martin Brandenburg
    Commented Nov 13 at 11:36
  • $\begingroup$ It's a question, in fact. $\endgroup$
    – Naïm Favier
    Commented Nov 13 at 14:19
  • $\begingroup$ The OP asked for a formalization of sub-xyz in categories. Your link answers it. $\endgroup$
    – Martin Brandenburg
    Commented Nov 13 at 17:36
  • $\begingroup$ @NaïmFavier: Thanks a lot. The way to look at subobjects in the category of monoids really helps. $\endgroup$
    – peach J
    Commented Nov 14 at 3:31

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