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There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.

My question is what characterizes this list?

Is there a general structure on a bicategory $B$ that we can develop a theory of $\mathcal{V}$-enriched category, for $\mathcal{V}\in ob B$ (along with base change functors) that "behave well"?

For example, if we take skew-closed categories as the enriching context, there is no yoneda lemma unless the enriching category is left normal. Therefore there should be some restrictions on the bicategory.

My motivation is that I work in one of these contexts (closed categories), and it seems the only property I really need in order to do 2-categorial constructions is the enriched yoneda lemma (the yoneda structure of $V$-categories). The details of the enriching context are not relevent to the construction, and can be removed, if we only assumed a bicategory where one can do enriched category theory.

EDIT: The definition be something as follows (I'm not being precise about what notion of $3$-categories we work in as I don't know what it should be):

There is a $3$-category called $Context$ (whose objects are bicategories of enrichment contexes, with $MonCat$ being the prime example) along with a $3$-functor $(-)Cat: Context \to 3Cat$ that satisfies a bunch of axioms that make it look like the theory of enrichment along with base change.

Then there should be a sub-$3$-category of contexes whose "enrichement theory functor $(-)Cat$" gives rise to $3$-categories equipped with a yoneda structure that is compatible along base change (and other structure?).

Aside: If this sub-$3$-category exists, there can be a "completeness functor" (left adjoint to the inclusion). And in this functor for example, the completeness of $MonCat$ would probably be $SymClosedCat$ if that makes sense. For $SkewClosedCat$ the closure should be the $ClosedCat$ If we have such a completeness functor, and if the subcategory contains an initial object (which I believe exists) then we can have a very thorough description of "what is enrichment" and when two bicategories give the same information when enriching over them.

Following the comment: if we manage to define this situation, then enrichement in $n$-categories can be described using descent of $(-)Cat$ along all the forgetful functors from $nCat$ to $(n-1)Cat$

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  • $\begingroup$ Is "${\cal V}\in Bicat$" a sufficient request? You can do quite a lot of category theory in bicategory-enriched categories. For example, you can re-do universal algebra and Gabriel-Ulmer duality. $\endgroup$
    – fosco
    Commented Aug 27, 2018 at 19:47
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    $\begingroup$ You may find Garner and Shulman's work interesting. They develop a theory of bicategories enriched in a tricategory, and show that for a certain monoidal bicategory they call $\mathcal F_\infty$, the passage from a locally cocomplete proarrow equipment to its proarrow equipment of enriched categories is a free cocompletion under a certain type of $\mathcal F_\infty$-enriched colimit called a "tight collage". I think if I were looking for a general context to talk about enrichment I would look for a similar "free cocompletion" type statement. $\endgroup$
    – Tim Campion
    Commented Aug 27, 2018 at 22:46
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    $\begingroup$ Mike Shulman’s paper Enriched indexed categories lays out a quite general setting for enriched cats, which might be relevant, though I haven’t time now to check through whether it covers all your examples. Martin Lundfall’s masters thesis Models of Linear Dependent Type Theory presents of a slightly less general but still quite general setup, in terms of linear logic. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Aug 28, 2018 at 6:46
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    $\begingroup$ Tom Leinster's Generalized Enrichment for Categories and Multicategories argues that the most general context in which one can enrich a $T$-multicategory is a $T'$-multicategory, where $T'$ is the monad whose algebras are $T$-multicategories. This leads to the conclusion that the most general thing in which to enrich a category is a virtual double category (which Leinster calls an "fc-multicategory"). $\endgroup$
    – Mike Shulman
    Commented Sep 5, 2018 at 17:38
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    $\begingroup$ It's worth noting that the question contains a counterexample to Leinster's claim: virtual double categories do not subsume skew-monoidal categories, yet categories may be enriched in skew-monoidal categories. $\endgroup$
    – varkor
    Commented Feb 9 at 10:55

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