Definition:Injective on Objects

Definition

Let $\mathbf C$ and $\mathbf D$ be metacategories.

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

Then $F$ is said to be injective on objects if and only if for all objects $C_1, C_2$ of $\mathbf C$:

$F C_1 = F C_2$ implies $C_1 = C_2$

Also see

• Definition:Surjective on Objects
• Definition:Injective on Morphisms
• Definition:Faithful Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.): Chapter $7$ Naturality: $\S 7.1$ Category of Categories: Definition $7.1$