nLab category enriched in a bicategory

Idea

The notion of category enriched in a bicategory is the many-object-generalization of the notion of an enriched category enriched in a monoidal category via regarding a monoidal category as a bicategory with a single object.

Originally Bénabou called these polyads.

Definition

Let $B$ be a bicategory, and write $\otimes$ for horizontal (1-cell) composition (written in Leibniz order). A category enriched in the bicategory $B$ consists of a set $X$ together with

• A function $p: X \to B_0$,
• A function $\hom: X \times X \to B_1$, satisfying the typing constraint $\hom(x, y): p(x) \to p(y)$,
• A function $\circ: X \times X \times X \to B_2$, satisfying the constraint $\circ_{x, y, z}: \hom(y, z) \otimes \hom(x, y) \to \hom(x, z)$,
• A function $j: X \to B_2$, satisfying the constraint $j_x: 1_{p(x)} \to \hom(x, x)$,

such that the associativity and unitality diagrams, as written above, commute. Viewing a monoidal category $M$ as a 1-object bicategory $\Sigma M$, the notion of enrichment in $M$ coincides with the notion of enrichment in the bicategory $\Sigma M$.

Equivalently this is simply a lax functor from the codiscrete category on $X$ into $B$. In particular if $X$ is the singleton set then this is the same as a monad.

If $X$, $Y$ are sets which come equipped with enrichments in $B$, then a $B$-functor consists of a function $f: X \to Y$ such that $p_Y \circ f = p_X$, together with a function $f_1: X \times X \to B_2$, satisfying the constraint $f_1(x, y): \hom_X(x, y) \to \hom_Y(f(x), f(y))$, and satisfying equations expressing coherence with the composition and unit data $\circ$, $j$ of $X$ and $Y$. (Diagram to be inserted, perhaps.)

Related concepts

• 2-category equipped with proarrows – Categories enriched in an equipment
• locally cocomplete bicategory

References

Free cocompletions of categories enriched in bicategories are constructed in section 9 of

• Nathanael Arkor, Dylan McDermott, Presheaves and cocompletions in formal category theory [arXiv:2604.22370]

Discussion of Kleisli objects (collages) for monads generalized to categories enriched in bicategories is in section 15.9 of

• Richard Garner, Michael Shulman, Enriched categories as a free cocompletion (arXiv:1301.3191)