2-category

Definition

A 2-category ${\mathcal {C}}$ is the following data:

• Objects: A collection $\operatorname {Ob} {\mathcal {C}}$ of objects.
• Morphisms: For any objects $A,B\in \operatorname {Ob} {\mathcal {C}}$, a collection ${\mathcal {C}}(A,B)$ of morphisms. Every element in ${\mathcal {C}}_{1}(A,B)$ 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 $f\in {\mathcal {C}}(A,B)$ is also written as $f:A\to B$. The collection ${\mathcal {C}}(A,B)$ is sometimes also denoted $\operatorname {Hom} _{\mathcal {C}}(A,B)$ or simply $\operatorname {Hom} (A,B)$.
• Identity morphism: For every object $A\in \operatorname {Ob} {\mathcal {C}}$, a distinguished morphism $\operatorname {id} _{A}\in {\mathcal {C}}(A,A)$. This is called the identity morphism of $A$.
• Composition rule: For $A,B,C\in \operatorname {Ob} {\mathcal {C}}$, a map, called composition of morphisms, from ${\mathcal {C}}(B,C)\times {\mathcal {C}}(A,B)$ to ${\mathcal {C}}(A,C)$. This map is denoted by $\circ$.
• 2-morphisms: For any two objects $A,B\in \operatorname {Ob} {\mathcal {C}}$ and any two morphisms $f,g\in {\mathcal {C}}_{1}(A,B)$, a collection ${\mathcal {C}}_{2}(f,g)$ of 2-morphisms from $f$ to $g$. If $\alpha$ is such a morphism, we write $\alpha :f\implies g$. The collection of such 2-morphisms may be denoted $\operatorname {Hom} (f,g)$.
• Identity 2-morphism: For any two objects $A,B\in \operatorname {Ob} {\mathcal {C}}$ and any morphism $f\in \operatorname {Hom} (A,B)$, a 2-morphism $\operatorname {id} _{f}\in \operatorname {Hom} (f,f)$.
• Composition of 2-morphisms: For any two objects $A,B$, and any three morphisms $f,g,h\in \operatorname {Hom} (A,B)$, a map $\operatorname {Hom} (g,h)\times \operatorname {Hom} (f,g)\to \operatorname {Hom} (f,h)$.
• 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 $\operatorname {Hom} (f_{1},f_{2})\times operatorname{Hom}(g_{1},g_{2})\to \operatorname {Hom} (f_{1}\circ g_{1},f_{2}\ circg_{2})$.

satisfying the following conditions:

• Associativity of composition of morphisms: For $A,B,C,D\in \operatorname {Ob} {\mathcal {C}}$, with $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C),h\in {\mathcal {C}}(C,D)$, we have $h\circ (g\circ f)=(h\circ g)\circ f$.
• Identity morphism behaves as an identity: For $A,B\in \operatorname {Ob} {\mathcal {C}}$, with $f\in {\mathcal {C}}(A,B)$, we have $f\circ \operatorname {id} _{A}=\operatorname {id} _{B}\circ f=f$.
• 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 ${\mathcal {C}}$ is a category (that we'll also call ${\mathcal {C}}$), along with, for every $A,B\in {\mathcal {C}}$, a category whose collection of objects is the collection of morphisms from $A$ to $B$, along with a horizontal composition...Fill this in later