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...

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...

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

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...

category generated by it. Since Hom S ′ {\displaystyle \operatorname {Hom} _{S'}} is a functor, ( x , y ) ↦ Sing ⁡ | Hom C ⁡ ( x , y ) | {\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...

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...

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...

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...

direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in...

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...

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...

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

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

{\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...

{\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...

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...

to make Rel a dagger category. The category has two functors into itself given by the hom functor: A binary relation R ⊆ A × B and its transpose RT ⊆...

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

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...

in C, the natural functor (evaluation at the identity) Hom _ ( h x , F ) → F ( x ) {\displaystyle {\underline {\operatorname {Hom} }}(h_{x},F)\to F(x)}...

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...

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...

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...

we have ∘ : hom ( b , c ) × hom ( a , b ) ↦ hom ( a , c ) {\displaystyle \circ :{\text{hom}}(b,c)\times {\text{hom}}(a,b)\mapsto {\text{hom}}(a,c)} The...

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

category, it is equivalent to require that the hom functor Hom C ⁡ ( − , Q ) {\displaystyle \operatorname {Hom} _{\mathbf {C} }(-,Q)} carries monomorphisms...