---- > [!definition] Definition. ([[faithful functor]]) > Let $\mathsf{C}$ and $\mathsf{D}$ be (locally small) [[category|categories]] and $\mathscr{F}:\mathsf{C} \to \mathsf{D}$ a [[covariant functor|functor]]. For each pair $A,B$ of objects, $\mathscr{F}$ defines a function $\text{Hom}_{\mathsf{C}}(A,B) \to \text{Hom}_{\mathsf{D}} \big( \mathscr{F}(A), \mathscr{F}(B) \big);$ if this function is [[injection|injective]], then we call $\mathscr{F}$ **faithful**. ^definition > [!basicexample] > Let $G$ be a [[group]], viewed as a one-object [[category]]. A [[covariant functor|functor]] $G \to \mathsf{Set}$ is characterized by the selection of a set $S$ and an assignment to each $g \in G$ a set-isomorphism $f \in \text{Perm}(X)$, i.e., a [[group action]]. A [[faithful group action]] precisely one where this map $G \to \text{Perm}(X)$ is an [[injection]]. The same notion applies also to [[faithful group representation]]. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```