nLab enriched bicategory

Idea

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.

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

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

  in $\mathcal{V}$

• for every object $x$ an identity morphism

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

  from the tensor unit of $\mathcal{V}$;

• a natural 2-ismorphism

  $$\alpha : comp(Id \otimes comp) \Rightarrow comp(comp \otimes Id)$$

  called the associator

• similarly left and right unitors

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

Examples

• When $\mathcal{V} = Cat$, 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.

References

Bicategories enriched in monoidal 2-categories first appeared in:

• Syméon Bozapalides, Théorie formelle des bicatégories, PhD thesis, 1976.

• Syméon Bozapalides, Bicatégories relatives à une 2-catégorie multiplicative, CRAS, 1976.

The definition appeared independently 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),

and in Chapter 7 of

• Alex Hoffnung, Foundations of Categorified Representation Theory (pdf)

and studied further in

• Richard Garner and Mike Shulman, Enriched categories as a free cocompletion, arXiv

A theorem about representing lax enriched functors as lax algebras:

• Cristina Pedicchio?, Relazioni tra K-morfismi e lax-algebre (1979), pdf