Definition:Embedding of Categories

Definition

Let $C$ and $D$ be categories.

Let $F : C \to D$ be a functor.

Definition 1

The functor $F$ is an embedding if and only if it is:

• injective on objects
• faithful

Definition 2

The functor $F$ is an embedding if and only if it is injective on morphisms.

Definition 3

The functor $F$ is an embedding if and only if it is a monomorphisms in the category of categories.

Also see

• Equivalence of Definitions of Embedding of Categories