One notion often called an enriched bicategory is the same as that of an enriched category, but where the enriching context $\mathcal{V}$ is allowed to be generalized from a monoidal category to a monoidal bicategory, while suitably weakening the associativity and unitality conditions on the enrichment. Thus, it has a collection of objects with hom-objects $C(x,y)\in\mathcal{V}$. It may also naturally be called a pseudo enriched category.
A different notion that is also sometimes called an enriched bicategory is that of a bicategory enriched over a monoidal 1-category $V$ (which must be at least braided) at the level of 2-cells only. Thus it has a collection of objects, with 1-morphisms between the objects, and for any parallel 1-morphisms $f,g\colon x\to y$, a hom-object $C(x,y)(f,g) \in V$. This can be identified with a $(V Cat)$-enriched bicategory in the previous sense, so on this page we focus on the former, more general, definition.
For $\mathcal{V}$ a monoidal bicategory, a $\mathcal{V}$-enriched (bi)category $C$ consists of
a collection of objects;
for every ordered pair $(x,y)$ of objects a hom-object $C(x,y) \in \mathcal{V}$
for every ordered triple $(x,y,z)$ a composition morphism of the form
in $\mathcal{V}$
for every object $x$ an identity morphism
from the tensor unit of $\mathcal{V}$;
called the associator
similarly left and right unitors
such that some more or less evident coherence conditions hold (see the references).
When $\mathcal{V} = Cat$, a $\mathcal{V}$-enriched bicategory is just a plain bicategory.
When $\mathcal{V}$ is an ordinary monoidal category, a $\mathcal{V}$-enriched bicategory is just an ordinary enriched category.
When $\mathcal{V}$ is the cartesian monoidal 2-category of bicategories, pseudo 2-functors, and icons, then a $\mathcal{V}$-enriched bicategory is an iconic tricategory?.
When $\mathcal{V}$ is the cartesian monoidal 2-category of fully faithful functors, then a $\mathcal{V}$-enriched bicategory is a weak F-category.
The definition first appeared in
A definition also appeared in
Forcey has studied the combinatorics of polytopes associated to enrichment and higher categories in detail. See for example
The definition is reviewed in
and in Chapter 7 of
and studied further in
Last revised on June 2, 2021 at 07:17:15. See the history of this page for a list of all contributions to it.