Profunctor

In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. A profunctor (also named distributor by the French school and module by the Sydney school) ϕ {displaystyle ,phi } from a category C {displaystyle C} to a category D {displaystyle D} , written is defined to be a functor where D o p {displaystyle D^{mathrm {op} }} denotes the opposite category of D {displaystyle D} and S e t {displaystyle mathbf {Set} } denotes the category of sets. Given morphisms f : d → d ′ , g : c → c ′ {displaystyle fcolon d o d',gcolon c o c'} respectively in D , C {displaystyle D,C} and an element x ∈ ϕ ( d ′ , c ) {displaystyle xin phi (d',c)} , we write x f ∈ ϕ ( d , c ) , g x ∈ ϕ ( d ′ , c ′ ) {displaystyle xfin phi (d,c),gxin phi (d',c')} to denote the actions. Using the cartesian closure of C a t {displaystyle mathbf {Cat} } , the category of small categories, the profunctor ϕ {displaystyle phi } can be seen as a functor where D ^ {displaystyle {hat {D}}} denotes the category S e t D o p {displaystyle mathrm {Set} ^{D^{mathrm {op} }}} of presheaves over D {displaystyle D} . A correspondence from C {displaystyle C} to D {displaystyle D} is a profunctor D ↛ C {displaystyle D rightarrow C} . The composite ψ ϕ {displaystyle psi phi } of two profunctors