more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure...

set with generator u. The forgetful functor Grp → Set on the category of groups is represented by (Z, 1). The forgetful functor Ring → Set on the category...

Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns...

of sets is a functor. Functors like these, which "forget" some structure, are termed forgetful functors. Another example is the functor Rng → Ab which...

algebraic forgetful functors. The free functor F : Set → Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and...

lift exists that is a final object among all lifts. For example, the forgetful functor QCoh → Sch {\displaystyle {\textrm {QCoh}}\to {\textrm {Sch}}} from...

category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor. Consider a category C of algebraic structures;...

category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties...

there are forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and addition, respectively. Both of these functors have left adjoints...

category of sets and functions) is a faithful functor. The functor U is to be thought of as a forgetful functor, which assigns to every object of C its "underlying...

morphisms are functions preserving this structure. There is a natural forgetful functor U : Top → Set to the category of sets which assigns to each topological...

an isomorphism. The forgetful functors in algebra, such as from Grp to Set, are conservative. More generally, every monadic functor is conservative. In...

is given by the product order on the cartesian product. We have a forgetful functor PreOrd → Set that assigns to each preordered set the underlying set...

there is a forgetful functor from Cat (the category of categories) to Quiv (the category of multidigraphs). Its left adjoint is a free functor which, from...

For example, the forgetful functor from the category of compact Hausdorff spaces to sets is monadic. However the forgetful functor from all topological...

homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category. The forgetful functor has a left adjoint (which associates...

the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing...

In various situations, free semilattices exist. For example, the forgetful functor from the category of join-semilattices (and their homomorphisms) to...

a functor from K {\displaystyle K} -Vect to K {\displaystyle K} -Alg. This means that T {\displaystyle T} is left adjoint to the forgetful functor U {\displaystyle...

that the composition of two left adjoint functors is also a left adjoint functor. Here, the forgetful functor from commutative algebras to vector spaces...

topological functor is one which has similar properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits...

object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal...

{\displaystyle \mathrm {Set} .} The forgetful functor U : T o p → S e t {\displaystyle U:\mathrm {Top} \to \mathrm {Set} } induces a functor U ¯ : C o n e ( Y ) → C...

In fact, a {\displaystyle a} is the left adjoint functor to the inclusion functor (or forgetful functor) from the category of sheaves to the category of...

lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets. Free complete lattices...

conservative functor A conservative functor is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from...

category. This is the internal hom [x, y]. Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom...

{\textbf {Set}}} is the forgetful functor, meaning R ( − ) {\displaystyle R^{(-)}} is a left adjoint of the forgetful functor. Many statements true for...

T} is the forgetful functor mapping an abelian group to its underlying set, and s {\displaystyle s} is some fixed set (regarded as a functor from 1), then...

can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor. More precisely, let C {\displaystyle {\mathcal {C}}}...