Definition 11.1.2.1.1 (00XD): Subcategories

11.1.2 Subcategories

Let $\mathcal{C}$ be a category.

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

1.
Objects. We have $\operatorname {\mathrm{Obj}}(\mathcal{A})\subset \operatorname {\mathrm{Obj}}(\mathcal{C})$.
2.
Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{A})$, we have

\[ \operatorname {\mathrm{Hom}}_{\mathcal{A}}(A,B) \subset \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B). \]
3.
Identities. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{A})$, we have

\[ \mathbb {1}^{\mathcal{A}}_{A} = \mathbb {1}^{\mathcal{C}}_{A}. \]
4.
Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}(\mathcal{A})$, we have

\[ \circ ^{\mathcal{A}}_{A,B,C} = \circ ^{\mathcal{C}}_{A,B,C}. \]

Definition 11.1.2.1.2 ▶ Full Subcategories

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 $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{A})$, the inclusion

\[ \iota _{A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{A}}(A,B)\hookrightarrow \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \]

is surjective (and thus bijective).

Definition 11.1.2.1.3 ▶ Strictly Full Subcategories

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 $\operatorname {\mathrm{Obj}}(\mathcal{A})$ is closed under isomorphisms.¹

¹That is, given $A\in \operatorname {\mathrm{Obj}}(\mathcal{A})$ and $C\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, if $C\cong A$, then $C\in \operatorname {\mathrm{Obj}}(\mathcal{A})$.

Definition 11.1.2.1.4 ▶ Wide Subcategories

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

¹Further Terminology: Also called lluf.