Forgetful functor

Definition

A forgetful functor is a type of functor in category theory that disregards certain structures or properties of the objects in a category while preserving the relationships between them. Essentially, it 'forgets' extra information about the objects but retains the basic structure needed to understand how they relate to each other. This is particularly useful in order theory, as it helps in simplifying complex structures to analyze their essential properties and relationships.

5 Must Know Facts For Your Next Test

1. Forgetful functors are commonly used in mathematics to relate different categories by simplifying structures, such as turning groups into sets by forgetting the group operation.
2. In order theory, forgetful functors can map ordered sets to sets, ignoring the order relations while still preserving the underlying set.
3. These functors often arise in conjunction with adjoint functors, where the forgetful functor is left adjoint to a free functor that adds structure back into the objects.
4. An example of a forgetful functor is the inclusion of vector spaces into sets, which forgets the vector space operations but retains the underlying set of vectors.
5. Forgetful functors help in understanding how more complex structures relate to simpler ones and are essential tools for establishing relationships in various mathematical contexts.

Review Questions

• How does a forgetful functor facilitate the study of relationships between different structures in mathematics?
  • A forgetful functor allows mathematicians to simplify complex structures by stripping away additional properties or operations while maintaining the essential relationships between objects. This simplification helps in analyzing how various structures relate to one another without getting bogged down by extraneous details. For instance, when studying ordered sets, forgetting the order enables one to focus solely on the set elements and their interactions.
• Discuss how forgetful functors interact with adjoint functors in category theory.
  • Forgetful functors often serve as left adjoints to free functors, creating a connection between two categories by mapping objects from one category to another while forgetting certain structures. For example, a forgetful functor might map a category of algebraic structures (like groups or rings) to the category of sets, while a corresponding free functor would take sets and equip them with an algebraic structure. This relationship illustrates how adding or removing structure can reveal insights into the nature of mathematical objects.
• Evaluate the significance of forgetful functors in order theory and provide examples of their applications.
  • Forgetful functors are significant in order theory because they simplify the analysis of ordered sets and posets by ignoring specific ordering information. For example, a forgetful functor can take an ordered set and treat it simply as a set, allowing researchers to explore set-theoretic properties without concern for order. Applications include finding homomorphisms between ordered structures or analyzing lattice properties without being constrained by their ordered nature, thus providing deeper insights into their fundamental characteristics.

Related terms

Functor: A functor is a map between categories that preserves the structure of categories by mapping objects to objects and morphisms to morphisms.

Adjunction: An adjunction is a pair of functors that establish a relationship between two categories, where one functor provides a way to go from one category to another while the other provides a way back.

Category: A category consists of objects and morphisms (arrows) between these objects, where the morphisms respect composition and identity.