hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and...

called a hom-functor. This functor is denoted: h A = H o m ( A , − ) {\displaystyle h_{A}=\mathrm {Hom} (A,-)} . The (covariant) hom-functor h A {\displaystyle...

the group homomorphism Hom(f, g): Hom(A2, B1) → Hom(A1, B2) is given by φ ↦ g ∘ φ ∘ f. See Hom functor. Representable functors We can generalize the previous...

tensor-hom adjunction is that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ⁡ ( X , − ) {\displaystyle \operatorname {Hom} (X,-)}...

from the fact the covariant Hom functor Hom(N, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits...

of sets. For each object A of C let Hom(A,–) be the hom functor that maps object X to the set Hom(A,X). A functor F : C → Set is said to be representable...

{\displaystyle \mathrm {hom} _{\mathcal {C}}(Fd,c)} , φ f {\displaystyle \varphi f} is the right adjunct of f {\displaystyle f} (p. 81). The functor F {\displaystyle...

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological...

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...

the contravariant hom-functor Hom(–, A) for some object A of C is called a representable presheaf. Some authors refer to a functor F : C o p → V {\displaystyle...

Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category...

theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two...

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

a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to...

functor is called the internal Hom functor, and the object A ⇒ B {\displaystyle A\Rightarrow B} is called the internal Hom of A {\displaystyle A} and B...

{\displaystyle \mathrm {Hom} (A\otimes B,C)\cong \mathrm {Hom} (A,B\Rightarrow C).} Here, Hom denotes the (external) Hom-functor of all morphisms in the...

category, enriched functor, etc... reduce to the original definitions from ordinary category theory. An enriched category with hom-objects from monoidal...

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...

exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from...

product and the Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext...

objects a and b the hom-set Hom ⁡ ( a , b ) {\displaystyle \operatorname {Hom} (a,b)} acquires a structure of a category as a functor category. A vertical...

internal tensor product internal Hom The functors f ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} form an adjoint functor pair, as do f ! {\displaystyle...

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...

polynomial functor if the following equivalent conditions hold: For every pair of vector spaces X, Y in V {\displaystyle {\mathcal {V}}} , the map F : Hom ⁡ (...

considering functors between two R-linear categories, one often restricts to those that are R-linear, so those that induce R-linear maps on each hom-set. Any...

reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those,...

France Hom, Šentrupert, a dispersed settlement in Slovenia Hom-e Khosrow, a village in Iran Hom bundle, in topology Hom functor, in category theory Hom ⁡ (...

{\displaystyle F:\mathrm {Hom} _{\mathbf {Vect} }(T,U)\rightarrow \mathrm {Hom} _{\mathbf {Vect} }(F(T),F(U)),} where Hom is notation for Hom functor. If this map...

called the free functor is a left adjoint to the faithful functor U; that is, there is a bijection Hom S e t ⁡ ( X , U ( B ) ) ≅ Hom C ⁡ ( F ( X ) , B...

computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality. The loop...