Horizontal Composition
Definition
Def Pasting Diagram Given a category
, a pasting diagram in is a sequence of composable morphisms:
Horizontal Composition
Given a pair of natural transformations
and as indicated in the diagram: there is a natural transformation
whose component at is defined as
Lemma
Lemma Middle Four Interchange Given functors and natural transformations
the natural transformation
defined by first composing vertically and then composing horizontally equals the natural transformation defined by first composing horizontally and then composing vertically: Proof
2-Category
Definition
Def 2-Category A 2-category is comprised of:
- A collection of
-cells, also known as objects. - A collection of
-cells, also known as -morphisms between pairs of objects: - A collection of
-cells, also known as -morphisms between parallel pairs of -morphisms : So that the following conditions hold:
- The objects and
-morphisms form a category, with identities . - For each fixed pair of objects
and , the -morphisms and -morphisms between such form a category under an operation called vertical composition with identities . - There is also a category whose objects are the objects in which a morphism from
to is a -cell under an operation called horizontal composition, with identities . - The horizontal composite
of identities for vertical composition must be the identity for the composite -morphisms. - The law of middle four interchange holds.
