Forgetful functor information

In mathematics, 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...

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

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...

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...

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...

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...

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

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

unique functor F' : C(G) → D such that U(F')∘I=F, i.e. the following diagram commutes: The functor C is left adjoint to the forgetful functor U. Mathematics...

itself is a forgetful functor. free functor A free functor is a left adjoint to a forgetful functor. For example, for a ring R, the functor that sends...

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

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...

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

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...

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...

homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure...

{\displaystyle R} is the ring of integers, then this is just the forgetful functor from modules to abelian groups. Extension of scalars changes R-modules...

recovered from the category of representations of it together with the forgetful functor to the category of vector spaces. The Grothendieck ring of the category...

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

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

free objects) is a functor from the category of sets to the category of groups. This functor is left adjoint to the forgetful functor from groups to sets...