:PROPERTIES: :ID: 89fa9e35-523b-4075-8f44-5f296f2586ef :mtime: 20240111104043 20240109120814 :ctime: 20240109120026 :END: #+title: Functor #+filetags: :public:project: * Definition of a Functor In the context of category theory, a *functor* is a type of structure-preserving transformation between two categories. Let $C$ and $D$ be 2 categories. We say that $F : C \longrightarrow D$ is a functor from $C$ to $D$ if has the 2 following structures 1) *object mapping*: A way of mapping the objects of $C$ to the objects of $D$, \[Fo : Obj_C \to Obj_D\] 2) *arrow mapping*: A way of mapping the arrows of $C$ to the arrows of $D$, so \[Fa : (A \rightsquigarrow_{C} B) \to (Fo(A) \rightsquigarrow_{D} Fo(B)) \] Subject to the 2 following conditions 1) *respects identity* : A functor must map identities to identities: \[Fa(id_{C}) = id_{D}\] 2) *respects composition*: A functor must distribute over composition \[Fa(f \gg_{ABC} g) = Fa(f) \gg_{Fo(A)Fo(B)Fo(C)} Fa(g)\]