PlanetMath: forgetful functor

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forgetful functor

(Definition)

Let $\mathcal{C}$ and $\mathcal{D}$ be categories such that each object of $\mathcal{C}$ can be regarded an object of $\mathcal{D}$ by suitably ignoring structures may have as a $\mathcal{C}$ -object but not a $\mathcal{D}$ -object. A functor $U:\mathcal{C} \to \mathcal{D}$ which operates on objects of $\mathcal{C}$ by “forgetting” any imposed mathematical structure is called a forgetful functor. The following are examples of forgetful functors:

$U:\mathbf{Grp} \to \mathbf{Set}$ takes groups into their underlying sets and group homomorphisms to set maps.
$U:\mathbf{Top} \to \mathbf{Set}$ takes topological spaces into their underlying sets and continuous maps to set maps.
$U:\mathbf{Ab} \to \mathbf{Grp}$ takes abelian groups to groups and acts as identity on arrows.

Forgetful functors are often instrumental in studying adjoint functors.

"forgetful functor" is owned by RevBobo.

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