2-Categories

Horizontal Composition

Definition

Def Pasting Diagram
Given a category $\mathbf{C}$, a pasting diagram in $\mathbf{C}$ is a sequence of composable morphisms:

Horizontal Composition

Given a pair of natural transformations $\alpha :F\to G$ and $\beta :H\to K$ as indicated in the diagram:

there is a natural transformation $\beta \ast \alpha :HF\to KG$ whose component at $C\in \text{obj}\mathbf{C}$ is defined as $(\beta \ast \alpha {)}_C={\beta }_{G(C)}\circ H({\alpha }_C)=K({\alpha }_C)\circ {\beta }_{F(C)}$

Lemma

Lemma Middle Four Interchange
Given functors and natural transformations

the natural transformation $JF\to LH$ 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 $0$-cells, also known as objects.
• A collection of $1$-cells, also known as $1$-morphisms between pairs of objects: $f:A\to B$
• A collection of $2$-cells, also known as $2$-morphisms between parallel pairs of $1$-morphisms $\alpha :f\Rightarrow g$:

So that the following conditions hold:

• The objects and $1$-morphisms form a category, with identities $1_C:C\to C$.
• For each fixed pair of objects $C$ and $D$, the $1$-morphisms $f:C\to D$ and $2$-morphisms between such form a category under an operation called vertical composition with identities $1_f:f\to f$.
• There is also a category whose objects are the objects in which a morphism from $C$ to $D$ is a $2$-cell $\alpha :f\Rightarrow g$ under an operation called horizontal composition, with identities $1_{1_C}$.
• The horizontal composite $1_h\ast 1_f$ of identities for vertical composition must be the identity $1_{h\circ f}$ for the composite $1$-morphisms.
• The law of middle four interchange holds.