I am confused about definition. If category $C$ is small and $F$ faithful functor such that $F:C\rightarrow SET$ where $F$ doesn't change the structure of $C$ then the pair $(C,F)$ is concrete category?
A concrete category has only forgetful functors?
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category-theory
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1What do you mean by "$F$ doesn't change the structure of $C$" ? – 2017-02-08
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2The title appears to claim that the only functor on a concrete category is a forgetful functor. This is not a true statement. The body says you are "confused about definition", but you do not clearly state what term you want help defining. – 2017-02-08
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0KonKan: if $C$ express abelian group and $F(C)$ hasn't the commutative property then structure changed. hardmath: i read from different sources the definition. the definition of concrete category has necessary faithful functor but not necessary forgetful? – 2017-02-08
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A concrete category is a category equipped with a faithful functor into $Set$. This usually can be thought of as a forgetful functor, since in cases like $Ab$ and $Ring$, the faithful functor $F$ just forgets the abelian group/ring structure. But the actual definition of a concrete category only involves the idea of a faithful functor.