n-categories enriched in (n+1)-categories

Question

Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a set of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an object $mor(a, b)$ in some symmetric monoidal category $S$. Composition of $n$-morphisms in $C$ takes the form of an $S$-morphism $mor(a, b)\otimes mor(b, c)\to mor(a, c)$.

I've come across some examples which seem to be well-described by the following generalization of enrichment. We replace the symmetric monoidal category $D$ with an $(n{+}1)$-category $D$. Each $k$-morphism of $C$ is assigned a $k$-morphism of $D$, for $0\le k < n$. (These assignments must of course satisfy various compatibility conditions. I'm only trying to sketch the rough idea here.) If $a$ and $b$ are compatible $(n{-}1)$-morphisms of $C$, $mor(a, b)$ is an $n$-morphism of $D$ (whose domain and range are determined my the assignments for $a$ and $b$). Composition of $n$-morphisms of $C$ takes the form of an $(n{+}1)$-morphism in $D$ from $mor(a, b) \bullet mor(b, c)$ to $mor(a, c)$, where $\bullet$ denotes composition of $n$-morphisms in $D$.

(To recover the more familiar symmetric-monoidal-category-enriched case, let $S$ be a symmetric monoidal category and let $D$ be the $(n{+}1)$-category whose $n$-morphisms are objects of $S$, whose $(n{+}1)$-morphisms are morphisms of $S$, and whose $k$-morphisms are trivial for $k < n$.)

My question is:

Does anyone know of references which discuss versions of enrichment similar to this, or which give natural examples of it?

All I've been able to find so far is Tom Leinster's article Generalized Enrichment for Categories and Multicategories, which discusses a very general version of $n$-categories enriched in $(n{+}1)$-categories. I'm not certain whether what I describe above can been seen as a special case of Leinster's definition.

Answer 1

The case $n=1$ is well-known in the category-theoretic literature and is called (of course) "categories enriched in a bicategory". It was introduced by Betti and Carboni (Cauchy-completion and the associated sheaf) and Walters (Sheaves and Cauchy-complete categories), and studied by Street (Enriched categories and cohomology) and other authors as well. Important examples include:

• If $S$ is a site and $D$ is the bicategory whose objects are those of $S$ and whose morphisms are sieves, then $D$-enriched categories that are both symmetric and Cauchy-complete can be identified with sheaves on $S$. See Walters' paper Sheaves on sites as Cauchy-complete categories.

• If $D$ is the bicategory of spans in a category $S$ with pullbacks, then $D$-enriched categories are very closely related to locally small fibrations over $S$. See Variation through Enrichment by all four authors mentioned above.

• If $D$ is the bicategory constructed from a monoidal fibration as in this paper of mine, then $D$-enriched categories are a sort of category that is "both indexed and enriched".

Personally, I think it is better in most cases to enrich in a double category than in a bicategory: in both examples above, $D$ underlies a double category. Recognizing this doesn't change the notion of "$D$-enriched category" but it does give a wider notion of $D$-enriched functor, which in particular makes the second example into an honest equivalence with locally small fibrations. There is a similar equivalence for the third example. (I don't know of anywhere that this is written down yet.)

Leinster's notion of "category enriched in an fc-multicategory" is a bit more general even than this: an fc-multicategory (which I prefer to call a virtual double category) is to a double category in the same way that a multicategory is to a monoidal category. I do know of some examples that require this full generality, but they are kind of contrived. For instance, if instead of a monoidal fibration you start with a "multicategorical fibration", then instead of a double category you get a virtual double category.

I'm not quite sure how you want to generalize this to $n>1$. One possibility would be to observe that when $D$ is a bicategory, a $D$-enriched category is the same as a lax functor into $D$ whose domain is the chaotic 1-category on some set of objects. But for $n$-categories with $n>2$, there is probably no unique notion of "lax functor"; you would have to choose at which level(s) you want it to be lax. I guess you might also choose different inputs to make the domain "chaotic on".

One version which is certainly useful is to keep the domain as a chaotic 1-category, and make the functors lax at the bottom level but pseudo everywhere above that. In some sense this is the most straightforward "categorification", which produces an obvious generalization of the notion of bicategory enriched in a monoidal bicategory. I'd be interested to hear what other examples you have in mind!