nLab
enriched bicategory

Contents

Idea

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

Definition

For 𝒱\mathcal{V} a monoidal bicategory, a 𝒱\mathcal{V}-enriched (bi)category CC consists of

  • a collection of objects;

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

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

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

    in 𝒱\mathcal{V}

  • for every object xx an identity morphism

    i x:IC(x,x) i_x : I \to C(x,x)

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

  • a natural 2-ismorphism

    α:comp(Idcomp)comp(compId) \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=Cat\mathcal{V} = Cat, a 𝒱\mathcal{V}-enriched bicategory is just a plain bicategory.

  • When 𝒱\mathcal{V} is an ordinary monoidal category, a 𝒱\mathcal{V}-enriched bicategory is just an ordinary enriched category.

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

  • When 𝒱\mathcal{V} is the cartesian monoidal 2-category of fully faithful functors, then a 𝒱\mathcal{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

Revised on September 11, 2011 04:03:55 by Mike Shulman (71.136.248.27)