Definition:Faithful Functor

Definition

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

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

Then $F$ is faithful if and only if for all objects $C_1, C_2$ of $\mathbf C$:

$F: \map {\operatorname{Hom}_{\mathbf C} } {C_1, C_2} \to \map {\operatorname{Hom}_{\mathbf D} } {F C_1, F C_2}, \ f \mapsto F f$

is an injection.

Here $\operatorname{Hom}$ signifies a hom class.

Also see

• Definition:Injective on Morphisms
• Definition:Full Functor
• Definition:Embedding of Categories

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.): Chapter $7$ Naturality: $\S 7.1$ Category of Categories: Definition $7.1$
• 2016: Emily Riehl: Category Theory in Context: Chapter $1$: Categories, Functors, Natural Transformations, $\S 1.5$: Equivalence of categories: Definition $1.5.7$