Contents

Idea

One notion often called an enriched bicategory is the same as that of an enriched category, but where the enriching context $𝒱$ 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 $C(x,y)\in 𝒱$. 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 $f,g:x\to y$, a hom-object $C(x,y)(f,g)\in V$. This can be identified with a $(V\mathrm{Cat})$-enriched bicategory in the previous sense, so on this page we focus on the former, more general, definition.

Definition

For $𝒱$ a monoidal bicategory, a $𝒱$-enriched (bi)category $C$ consists of

• a collection of objects;

• for every ordered pair $(x,y)$ of objects a hom-object $C(x,y)\in 𝒱$

• for every ordered triple $(x,y,z)$ a composition morphism of the form

  $${\mathrm{comp}}_{x,y,z}:C(x,y)\otimes C(y,z)\to C(x,z)$$comp_{x,y,z} : C(x,y)\otimes C(y,z) \to C(x,z)

  in $𝒱$

• for every object $x$ an identity morphism

  $$i_x:I\to C(x,x)$$i_x : I \to C(x,x)

  from the tensor unit of $𝒱$;

• a natural 2-ismorphism

  $$\alpha :\mathrm{comp}(\mathrm{Id}\otimes \mathrm{comp})\Rightarrow \mathrm{comp}(\mathrm{comp}\otimes \mathrm{Id})$$\alpha : comp(Id \otimes comp) \Rightarrow comp(comp \otimes Id)

  called the associator

• similarly left and right unitors

• such that some more or less evident coherence conditions hold (see the references).

Examples

• When $𝒱=\mathrm{Cat}$, a $𝒱$-enriched bicategory is just a plain bicategory.

• When $𝒱$ is an ordinary monoidal category, a $𝒱$-enriched bicategory is just an ordinary enriched category.

• When $𝒱$ is the cartesian monoidal 2-category of bicategories, pseudo 2-functors, and icons, then a $𝒱$-enriched bicategory is an iconic tricategory?.

• When $𝒱$ is the cartesian monoidal 2-category of fully faithful functors, then a $𝒱$-enriched bicategory is a weak F-category.

References

The definition first appeared in

• S. M. Carmody, Cobordism Categories , PhD thesis, University of Cambridge, 1995.

A definition also appeared in

• Steve Lack, The algebra of distributive and extensive categories , PhD thesis, University of Cambridge, 1995.

Forcey has studied the combinatorics of polytopes associated to enrichment and higher categories in detail. See for example

• S. Forcey, Quotients of the Multiplihedron as Categoried Associahedra , (arXiv:0803.2694).

The definition is reviewed in

• Alex Hoffnung, Notes on enrichment (pdf)