2-category

Definition

A 2-category $C$ is the following data:

Objects: A collection $O b C$ of objects.
Morphisms: For any objects $A, B \in O b C$ , a collection $C (A, B)$ of morphisms. Every element in $C_{1} (A, B)$ is termed a morphism from $A$ (i.e., with source or domain $A$ ) to $B$ (i.e., with target or co-domain $B$ ). The morphism sets for different pairs of objects are disjoint. Note that $f \in C (A, B)$ is also written as $f : A \to B$ . The collection $C (A, B)$ is sometimes also denoted $H_{o m C} (A, B)$ or simply $H o m (A, B)$ .
Identity morphism: For every object $A \in O b C$ , a distinguished morphism $i_{d A} \in C (A, A)$ . This is called the identity morphism of $A$ .
Composition rule: For $A, B, C \in O b C$ , a map, called composition of morphisms, from $C (B, C) \times C (A, B)$ to $C (A, C)$ . This map is denoted by $\circ$ .
2-morphisms: For any two objects $A, B \in O b C$ and any two morphisms $f, g \in C_{1} (A, B)$ , a collection $C_{2} (f, g)$ of 2-morphisms from $f$ to $g$ . If $α$ is such a morphism, we write $α : f ⟹ g$ . The collection of such 2-morphisms may be denoted $H o m (f, g)$ .
Identity 2-morphism: For any two objects $A, B \in O b C$ and any morphism $f \in H o m (A, B)$ , a 2-morphism $i_{d f} \in H o m (f, f)$ .
Composition of 2-morphisms: For any two objects $A, B$ , and any three morphisms $f, g, h \in H o m (A, B)$ , a map $H o m (g, h) \times H o m (f, g) \to H o m (f, h)$ .
Horizontal composition of 2-morphisms: For any three objects $A, B, C$ , with morphisms $f_{1}, f_{2} : A \to B$ and morphisms $g_{1}, g_{2} : B \to C$ , an operator $H o m (f_{1}, f_{2}) \times o p e r a t o r n a m e H o m (g_{1}, g_{2}) \to H o m (f_{1} \circ g_{1}, f_{2} c i r c g_{2})$ .

satisfying the following conditions:

Associativity of composition of morphisms: For $A, B, C, D \in O b C$ , with $f \in C (A, B), g \in C (B, C), h \in C (C, D)$ , we have $h \circ (g \circ f) = (h \circ g) \circ f$ .
Identity morphism behaves as an identity: For $A, B \in O b C$ , with $f \in C (A, B)$ , we have $f \circ i_{d A} = i_{d B} \circ f = f$ .
Associativity of composition of 2-morphisms
Identity 2-morphism behaves as an identity
Associativity of horizontal composition

Definition in terms of category definition

A 2-category $C$ is a category (that we'll also call $C$ ), along with, for every $A, B \in C$ , a category whose collection of objects is the collection of morphisms from $A$ to $B$ , along with a horizontal composition...Fill this in later