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The notion of an enriched category and that of a graded category are both similar in the sense that they both endow the usual morphisms of a category with additional structure.

A natural question is thus: Are these notions related in some way? For instance, for every category $C$ enriched over $V$, is there a graded category $(D,F)$ with ``essentially the same structure''? (presumably, this should be some form of categorical equivalence) What about the converse question?

Have these sorts of questions even been considered in the literature? The whole notion of a graded category seems to be a relatively obscure one, and there isn't even an n-cat lab article on the subject, so I've had a hard time looking through the literature to try to find answers, and thus would appreciate any insights into this matter.

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  • $\begingroup$ Without additional assumptions, the category of $\cal A$-graded categories seems to be isomorphic to the slice category ${\bf Cat}_{/\cal A}$ of functors ${\cal C} \to{\cal A}$ and commutative triangles. Now, you have two problems: for sure you have a forgetful functor ${\bf Cat}_{/\cal A} \to \bf Cat$, but a converse would mean that you have a canonical way to give any object a morphism to $\cal A$; there is no such way. Second, the slice construction doesn't always behave nicely in enriched category theory. $\endgroup$ – Fosco Loregian Jul 27 '17 at 9:33

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