Definition 2.2.0.1. A strict $2$-category $\operatorname{\mathcal{C}}$ consists of the following data:
A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
For every pair of objects $X, Y \in \operatorname{\mathcal{C}}$, a category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y)$. We refer to objects $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $1$-morphisms from $X$ to $Y$ and write $f: X \rightarrow Y$ to indicate that $f$ is a $1$-morphism from $X$ to $Y$. Given a pair of $1$-morphisms $f,g \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, we refer to morphisms from $f$ to $g$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $2$-morphisms from $f$ to $g$.
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, a composition functor
\[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z). \]For every object $X \in \operatorname{\mathcal{C}}$, an identity $1$-morphism $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$.
These data are required to satisfy the following conditions:
- $(1)$
For each object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism $\operatorname{id}_{X}$ is a unit for both right and left composition. That is, for every object $Y \in \operatorname{\mathcal{C}}$, the functors
\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto f \circ \operatorname{id}_{X} \]\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,X) \quad \quad g \mapsto \operatorname{id}_{X} \circ g \]are both equal to the identity.
- $(2)$
The composition law of $\operatorname{\mathcal{C}}$ is strictly associative. That is, for every quadruple of objects $W, X, Y, Z \in \operatorname{\mathcal{C}}$, the diagram of categories
\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\operatorname{id}\times \circ } \ar [d]^{\circ \times \operatorname{id}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Y) \ar [d]^{ \circ } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Z) } \]commutes (in the ordinary category $\operatorname{Cat}$).