The Clowder Project

A subcategory of $\mathcal{C}$ is a category $\mathcal{A}$ satisfying the following conditions:

1. Objects. We have $\text{Obj}\webleft (\mathcal{A}\webright )\subset \text{Obj}\webleft (\mathcal{C}\webright )$.
2. Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
   \[ \textup{Hom}_{\mathcal{A}}\webleft (A,B\webright ) \subset \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ). \]
3. Identities. For each $A\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
   \[ \mathbb {1}^{\mathcal{A}}_{A} = \mathbb {1}^{\mathcal{C}}_{A}. \]
4. Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
   \[ \circ ^{\mathcal{A}}_{A,B,C} = \circ ^{\mathcal{C}}_{A,B,C}. \]

A subcategory $\mathcal{A}$ of $\mathcal{C}$ is full if the canonical inclusion functor $\mathcal{A}\to \mathcal{C}$ is full, i.e. if, for each $A,B\in \text{Obj}\webleft (\mathcal{A}\webright )$, the inclusion

is surjective (and thus bijective).

A subcategory $\mathcal{A}$ of a category $\mathcal{C}$ is strictly full if it satisfies the following conditions:

1. Fullness. The subcategory $\mathcal{A}$ is full.
2. Closedness Under Isomorphisms. The class $\text{Obj}\webleft (\mathcal{A}\webright )$ is closed under isomorphisms.^[1]

A subcategory $\mathcal{A}$ of $\mathcal{C}$ is wide^[2] if $\text{Obj}\webleft (\mathcal{A}\webright )=\text{Obj}\webleft (\mathcal{C}\webright )$.