# nLab framed bicategory

### Context

#### 2-Category theory

2-category theory

## Definitions

• 2-category

• strict 2-category

• bicategory

• enriched bicategory

• ## Transfors between 2-categories

• 2-functor

• 2-natural transformation

• modification

• Yoneda lemma for bicategories

• ## Morphisms in 2-categories

• fully faithful morphism

• faithful morphism

• conservative morphism

• pseudomonic morphism

• discrete morphism

• eso morphism

• ## Structures in 2-categories

• mate

• cartesian object

• fibration in a 2-category

• codiscrete cofibration

• ## Limits in 2-categories

• 2-limit

• 2-pullback

• comma object

• inserter

• inverter

• equifier

• ## Structures on 2-categories

• monoidal 2-category

• Gray tensor product

• proarrow equipment

• # Contents

## Idea

For large classes of examples of bicategories the 1-morphisms naturally are of one of two different types:

1. special morphisms that may be taken to compose strictly among themselves;

2. more general morphisms that behave like bimodules.

The archetypical example is indeed the bicategory whose objects are algebras, and whose 1-morphisms are bimodules between these: every ordinary algebra homomorphism $f : A \to B$ induces an $A$-$B$-bimodule ${}_f B$ and this operation induces a 2-functor from the category of algebras and algebra homomorphisms into the bicategory of algebras and bimodules, which is the identity on objects.

For usefully working with bicategories of this kind, it is typically of crucial importance to remember this extra information. A framing on a bicategory is a way to encode this.

This is essentially the same as a proarrow equipment on a bicategory. See there for more.

## References

Last revised on April 11, 2016 at 15:45:43. See the history of this page for a list of all contributions to it.