underlying functor

A forgetful functor (also called underlying functor) is defined from a category of algebraic gadgets (groups, Abelian groups, modules, rings, vector spaces, etc.) to the category of sets. A forgetful functor leaves the objects and the arrows as they are, except for the fact they are finally considered only as sets and maps, regardless of their algebraic properties. Other forgetful functors neglect only part of the algebraic properties, e.g., the commutative law when passing from Abelian groups to groups, or multiplication when passing from rings to Abelian groups. Forgetful functors are covariant and faithful.