2-category

Definition

Definition as a category enriched over categories

A 2-category is an category enriched over the category of categories (in particular cases, it is usually enough to enrich over the category of locally small categories, which presents fewer foundational hassles).

Raw definition

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

• Objects: A collection $\mathrm{O}\mathrm{b}\mathcal{C}$ of objects.
• Morphisms: For any objects $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 ${\mathrm{H}}_{\mathrm{o}}(A,B)$ or simply $\mathrm{H}\mathrm{o}\mathrm{m}(A,B)$.
• Identity morphism: For every object $A\in \mathrm{O}\mathrm{b}\mathcal{C}$, a distinguished morphism ${\mathrm{i}}_{\mathrm{d}}\in \mathcal{C}(A,A)$. This is called the identity morphism of $A$.
• Composition rule: For $A,B,C\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$ 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 $\alpha :f⟹g$. The collection of such 2-morphisms may be denoted $\mathrm{H}\mathrm{o}\mathrm{m}(f,g)$.
• Identity 2-morphism: For any two objects $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$ and any morphism $f\in \mathrm{H}\mathrm{o}\mathrm{m}(A,B)$, a 2-morphism ${\mathrm{i}}_{\mathrm{d}}\in \mathrm{H}\mathrm{o}\mathrm{m}(f,f)$.
• Composition of 2-morphisms: For any two objects $A,B$, and any three morphisms $f,g,h\in \mathrm{H}\mathrm{o}\mathrm{m}(A,B)$, a map $\mathrm{H}\mathrm{o}\mathrm{m}(g,h)\times \mathrm{H}\mathrm{o}\mathrm{m}(f,g)\to \mathrm{H}\mathrm{o}\mathrm{m}(f,h)$.
• 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 $\mathrm{H}\mathrm{o}\mathrm{m}(f_1,f_2)\times operatornameHom(g_1,g_2)\to \mathrm{H}\mathrm{o}\mathrm{m}(f_1\circ g_1,f_{2}circ{g}_2)$.

satisfying the following conditions:

• Associativity of composition of morphisms: For $A,B,C,D\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, with $f\in \mathcal{C}(A,B)$, we have $f\circ {\mathrm{i}}_{\mathrm{d}}={\mathrm{i}}_{\mathrm{d}}\circ f=f$.
• 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