Forgetful functor

A functor that maps structured objects to underlying objects by discarding some structure while preserving the maps between them. It formalizes how one category’s objects can be viewed in a less structured category.

Functor: A mapping between categories that assigns objects and morphisms while preserving identities and composition. A forgetful functor is a particular kind of functor, obeying the same identity and composition laws.
Forgetful functor from groups to sets: The functor sending each group to its underlying set and each group homomorphism to its underlying function. It discards multiplication and identity data while retaining the elements and functions.
Category: A mathematical structure consisting of objects, morphisms between objects, identity morphisms, and associative composition. A forgetful functor compares two categories by mapping their objects and morphisms.
Free object: An object generated from specified data subject only to the relations required by a universal property. Free objects add structure to raw data, complementing the direction of a forgetful functor.
Underlying set: The set obtained from a structured object by retaining its elements and discarding additional operations or relations. Many forgetful functors send algebraic structures to their sets of elements.
Forgetful functor from rings to abelian groups: The functor sending each ring to its additive abelian group and each ring homomorphism to its additive map. It forgets multiplication while preserving the additive structure and compatible maps.
Morphism: A structure-preserving map between objects in a category. A forgetful functor sends each structured map to a map in the less structured category.
Forgetful functor to pointed sets: A functor that sends a structured object with a distinguished element to its underlying pointed set. It illustrates that forgetting can retain selected structure, such as a distinguished point.
Faithful functor: A functor that is injective on each set of morphisms between a fixed pair of objects. Common forgetful functors preserve distinct structure-preserving maps, making them faithful.
Forgetful functor from topological spaces to sets: The functor sending each topological space to its underlying set and each continuous map to its underlying function. It removes the topology but retains the points and continuous maps.

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