The Clowder Project

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

1. 1.

   Objects. We have $\operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )\subset \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.

2. 2.

   Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, we have

   \[ \operatorname {\mathrm{Hom}}_{\mathcal{A}}\webleft (A,B\webright ) \subset \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ). \]
3. 3.

   Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, we have

   \[ \mathbb {1}^{\mathcal{A}}_{A} = \mathbb {1}^{\mathcal{C}}_{A}. \]
4. 4.

   Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, 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 $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, 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. 1.

   Fullness. The subcategory $\mathcal{A}$ is full.

2. 2.

   Closedness Under Isomorphisms. The class $\operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$ is closed under isomorphisms.¹

A subcategory $\mathcal{A}$ of $\mathcal{C}$ is wide¹ if $\operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )=\operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.