Vergessungsfunktor

Vergessungsfunktor (German for “forgetful functor”) is a specific type of functor in category theory that maps objects and morphisms from one category to another by discarding some of the structure present in the source category. Unlike general functors, a Vergessungsfunktor preserves the underlying set or underlying structure that the target category does not require, while ignoring extra properties such as operations, relations, or topologies.

For example, the forgetful functor from the category of groups to the category of sets sends each group to its underlying set and each group homomorphism to the same map viewed as a function between sets. Similarly, a forgetful functor from the category of topological spaces to sets preserves only the point set, ignoring the topology, and a forgetful functor from the category of vector spaces over a fixed field to the category of abelian groups forgets the scalar multiplication while retaining the additive group structure.

Vergessungsfunktoren are usually not invertible; there is generally no way to recover the lost structure from the image. However, they often admit adjoints: the left adjoint of a forgetful functor is called a free functor (e.g., the free group functor left adjoint to the forgetful functor from groups to sets). In some contexts, a right adjoint may exist as well, providing a cofree construction.

The concept is central to the study of algebraic structures and their relationships because it formalises the process of simplifying a structure while maintaining essential compositional aspects. It also underlies many fundamental results, such as the existence of universal constructions and the transfer of properties between categories.