The Clowder Project

A category $\smash {\webleft (\mathcal{C},\circ ^{\mathcal{C}},\mathbb {1}^{\mathcal{C}}\webright )}$ consists of:

• Objects. A class $\text{Obj}\webleft (\mathcal{C}\webright )$ of objects.
• Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, a class $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$, called the class of morphisms of $\mathcal{C}$ from $A$ to $B$.
• Identities. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets
  \[ \mathbb {1}^{\mathcal{C}}_{A}\colon \text{pt}\to \textup{Hom}_{\mathcal{C}}\webleft (A,A\webright ), \]

  called the unit map of $\mathcal{C}$ at $A$, determining a morphism

  \[ \text{id}_{A} \colon A \to A \]

  of $\mathcal{C}$, called the identity morphism of $A$.

• Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets
  \[ \circ ^{\mathcal{C}}_{A,B,C} \colon \textup{Hom}_{\mathcal{C}}\webleft (B,C\webright )\times \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{C}}\webleft (A,C\webright ), \]

  called the composition map of $\mathcal{C}$ at $\webleft (A,B,C\webright )$.

1. Associativity. The diagram
   
   commutes, i.e. for each composable triple $\webleft (f,g,h\webright )$ of morphisms of $\mathcal{C}$, we have
   \[ \webleft (f\circ g\webright )\circ h = f\circ \webleft (g\circ h\webright ). \]
2. Left Unitality. The diagram
   

   commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have

   \[ \text{id}_{B}\circ f=f. \]
3. Right Unitality. The diagram
   

   commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have

   \[ f\circ \text{id}_{A}=f. \]