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 studied further in
Last revised on January 4, 2016 at 15:06:39. See the history of this page for a list of all contributions to it.