Vergessungsfunktoren

Vergessungsfunktoren, sometimes translated as forgetting functors, are a concept in category theory. They are functors that map from a category of structured objects to a less structured category, effectively "forgetting" some of the structure. For instance, a common example is the forgetful functor from the category of vector spaces over a field F to the category of sets. This functor maps a vector space to its underlying set of vectors, discarding the vector space operations (addition and scalar multiplication). Similarly, a forgetful functor can exist between categories of algebraic structures, such as groups, rings, or modules, where it maps an object to its underlying set and removes the associated operations.

The primary utility of forgetful functors lies in their ability to simplify problems. By moving to a less structured category, one can sometimes leverage simpler tools or theorems. For example, proving a property about vector spaces might be easier by first proving it about the underlying sets and then showing it is preserved by the relevant operations. Conversely, forgetful functors can also be used in conjunction with their adjoints (if they exist) to construct structures. The existence and properties of adjoint functors are central to many areas of mathematics and computer science, and forgetful functors often play a role in these constructions. They are fundamental in understanding the relationships between different mathematical categories.