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PROFUNCTOR

Profunctor

Generalization in mathematics

theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. A profunctor (also named distributor by the French

Profunctor

Correspondence

Topics referred to by the same term

between two polygons Correspondence (category theory), the opposite of a profunctor Correspondence (von Neumann algebra) or bimodule, a type of Hilbert space

Correspondence

Hom functor

Functor mapping hom objects to an underlying category

corepresentable. Note that Hom(–, –) : Cop × C → Set is a profunctor, and, specifically, it is the identity profunctor id C : C ↛ C {\displaystyle \operatorname {id}

Hom functor

Hom_functor

Presheaf (category theory)

Contravariant functor to Set

functor into C ^ {\displaystyle {\widehat {C}}} is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor

Presheaf (category theory)

Presheaf_(category_theory)

Functor

Mapping between categories

types to functions between some new types. Mathematics portal Anafunctor Profunctor Functor category Kan extension Pseudofunctor Mac Lane, Saunders (1971)

Functor

Yoneda lemma

Embedding of categories into functor categories

Guillaume; Gibbons, Jeremy (2018-07-30). "What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl)". Proc. ACM Program. Lang

Yoneda lemma

Yoneda_lemma

Distributor (disambiguation)

Topics referred to by the same term

distributor Distributor (category theory) in category theory, also known as a profunctor Distributor road, a road which serves to move traffic from local streets

Distributor (disambiguation)

Distributor_(disambiguation)

Arrow (computer science)

Much of the utility of arrows is subsumed by more general classes like profunctor (which requires only pre- and postcomposition with functions), which have

Arrow (computer science)

Arrow_(computer_science)

Denotational semantics

Study of programming languages via mathematical objects

for CSP). Recently, Winskel and others have proposed the category of profunctors as a domain theory for concurrency. State (such as a heap) and simple

Denotational semantics

Denotational_semantics

Isbell duality

Adjunction between a category of co/presheaf under the co/Yoneda embedding

{Lan_{z}y} } . Kan extension Limit (category theory) Isbell completion Profunctor (Baez 2022) (Di Liberti 2020, 2. Isbell duality) (Lawvere 1986, p. 169)

Isbell duality

Isbell_duality

Timeline of category theory and related mathematics

History of maths

representability 1973 Jean Bénabou Distributors (also called modules, profunctors, directed bridges) 1973 Pierre Deligne Proves the last of the Weil conjectures

Timeline of category theory and related mathematics

Timeline_of_category_theory_and_related_mathematics

Anafunctor

Mathematical notion

x , y ( f ) {\displaystyle F_{x,z}(gf)=F_{y,z}(g)\circ F_{x,y}(f)} . Profunctor The etymology of anafunctor is an analogy of the biological terms anaphase/prophase

Anafunctor

Bimodule

Abelian group equipped with compatible ring action on both sides

Tor. Profunctors can be seen as a categorical generalization of bimodules. Note that bimodules are not at all related to bialgebras. Profunctor Street

Bimodule

Glossary of category theory

objects and morphisms a , b {\displaystyle a,b} . profunctor Given categories C and D, a profunctor (or a distributor) from C to D is a functor of the

Glossary of category theory

Glossary_of_category_theory