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I'd like to know whether there exists a common language that encompasses both categories and enriched categories, so that results pertaining to either may be proven in a uniform way. I'd prefer it if that common language were a generalisation of both categories and enriched categories that is as far-reaching as possible.

Thank you very much in advance.

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closed as too broad by Pece, Song, Delta-u, José Carlos Santos, Parcly Taxel Feb 28 at 13:48

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Well, I'd say that you first have to have a notion of monoidal category in order to do any enriching. That said, once you've defined enriched categories (using this notion) one can prove things for both enriched categories and ordinary categories at the same time by regarding ordinary categories as being $\mathbf{Set}$ enriched. $\endgroup$ – jgon Feb 27 at 14:44
  • $\begingroup$ As for your second question, I think it's unclear what exactly you mean. That said, I think the answer is yes, but addressing all generalisations of ordinary categories is far too broad a question, and is not appropriate for MSE. Even asking what all the generalisations of ordinary categories are is probably too broad. $\endgroup$ – jgon Feb 27 at 14:48
  • $\begingroup$ @jgon Could you point to a rule of this site that would support your claim? Please don't take offence - I'm simply insufficiently familiar with this site. $\endgroup$ – AlgebraicsAnonymous Feb 27 at 15:54
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    $\begingroup$ @AlgebraicsAnonymous The first few points here math.stackexchange.com/help/dont-ask and the Be Specific section of math.stackexchange.com/help/how-to-ask as well as Too Broad close reason which states to "limit [the question] to a specific problem with enough detail to identify an adequate answer" and to "avoid asking multiple distinct questions at once". $\endgroup$ – Derek Elkins Feb 27 at 19:01
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    $\begingroup$ jgon already answered your question. Enriched category theory is a generalization of category theory, so this generalizes both enriched categories and $Set$-enriched (=traditional) categories. (You're basically asking for a 'common upper bound' of certain $A$ and $B$ where $A\le B$, then an obvious answer is $B$.) $\endgroup$ – Berci Feb 28 at 23:01