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A [[Functor|functor]] which is  "injective on Hom-sets" . Explicitly, a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381601.png" /> is called faithful if, given any two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381602.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381603.png" /> with the same domain and codomain, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381604.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381605.png" />. The name derives from the representation theory of groups: a permutation (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381606.png" />-linear) representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381607.png" /> is faithful if and only if it is faithful when considered as a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381608.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f0381609.png" />). A faithful functor reflects monomorphisms (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f03816010.png" /> monic implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f03816011.png" /> monic) and epimorphisms; hence if the domain category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038160/f03816012.png" /> is balanced (i.e. has the property that any morphism which is both monic and epic is an isomorphism) then it also reflects isomorphisms. A functor with the latter property is generally called conservative; however, some authors include this condition in the definition of faithfulness.
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A [[Functor|functor]] which is  "injective on Hom-sets" . Explicitly, a functor $  F : \mathfrak C \rightarrow \mathfrak D $
 +
is called faithful if, given any two morphisms $  \alpha , \beta : A \rightarrow B $
 +
in $  \mathfrak C $
 +
with the same domain and codomain, the equation $  F \alpha = F \beta $
 +
implies $  \alpha = \beta $.  
 +
The name derives from the representation theory of groups: a permutation (respectively, $  R $-
 +
linear) representation of a group $  G $
 +
is faithful if and only if it is faithful when considered as a functor $  G \rightarrow  \mathop{\rm Set} $(
 +
respectively $  G \rightarrow  \mathop{\rm Mod} _ {R} $).  
 +
A faithful functor reflects monomorphisms (that is, $  F \alpha $
 +
monic implies $  \alpha $
 +
monic) and epimorphisms; hence if the domain category $  \mathfrak C $
 +
is balanced (i.e. has the property that any morphism which is both monic and epic is an isomorphism) then it also reflects isomorphisms. A functor with the latter property is generally called conservative; however, some authors include this condition in the definition of faithfulness.
  
 
In Russian literature there seems to be some confusion between the terms  "faithful functor"  and  "exact functor" , see also [[Exact functor|Exact functor]].
 
In Russian literature there seems to be some confusion between the terms  "faithful functor"  and  "exact functor" , see also [[Exact functor|Exact functor]].

Revision as of 19:38, 5 June 2020


A functor which is "injective on Hom-sets" . Explicitly, a functor $ F : \mathfrak C \rightarrow \mathfrak D $ is called faithful if, given any two morphisms $ \alpha , \beta : A \rightarrow B $ in $ \mathfrak C $ with the same domain and codomain, the equation $ F \alpha = F \beta $ implies $ \alpha = \beta $. The name derives from the representation theory of groups: a permutation (respectively, $ R $- linear) representation of a group $ G $ is faithful if and only if it is faithful when considered as a functor $ G \rightarrow \mathop{\rm Set} $( respectively $ G \rightarrow \mathop{\rm Mod} _ {R} $). A faithful functor reflects monomorphisms (that is, $ F \alpha $ monic implies $ \alpha $ monic) and epimorphisms; hence if the domain category $ \mathfrak C $ is balanced (i.e. has the property that any morphism which is both monic and epic is an isomorphism) then it also reflects isomorphisms. A functor with the latter property is generally called conservative; however, some authors include this condition in the definition of faithfulness.

In Russian literature there seems to be some confusion between the terms "faithful functor" and "exact functor" , see also Exact functor.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
Faithful functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Faithful_functor&oldid=46902