nLab dagger functor

functors

\dagger functors

Idea

A map between two dagger categories that preserves the dagger category structure.

Definition

Definition

Given two \dagger-categories AA and BB, a \dagger-functor F:ABF : A \to B consists of a function F 0:Ob(A)Ob(B)F_0 : Ob(A) \to Ob(B) with a function F a,b:Hom A(a,b)Hom B(Fa,Fb)F_{a,b}:Hom_A(a,b) \to Hom_B(F a,F b) for every object a,b:Ob(A)a,b:Ob(A), where F a,bF_{a,b} is generally also denoted as FF, such that

  • for every object aOb(A)a \in Ob(A), F(1 a)=1 FaF(1_a)=1_{F a},
  • for every object a,b,cOb(A)a,b,c \in Ob(A) and morphisms fHom A(a,b)f \in Hom_A(a,b) and gHom A(b,c)g \in Hom_A(b,c), F(g Af)=Fg BFfF(g \circ_A f) = F g \circ_B F f
  • for every object a,bOb(A)a,b \in Ob(A) and morphism fHom A(a,b)f \in Hom_A(a,b), F(f A)=(Ff) BF(f^{\dagger_A}) = (F f)^{\dagger_B}.

There is another definition which violates the principle of equivalence, since the definition of the dagger in a dagger category in this case is a functor that imposes equations on objects: A \dagger-functor F:(C,)(D,)F : (C, \dagger) \to (D, \ddagger) between two \dagger-categories (C,)(C, \dagger) and (D,)(D, \ddagger) is a functor F:CDF : C \to D of the underlying categories, which commutes with the \dagger-structures in that F=F opF \circ \dagger = \ddagger \circ F^{op}.

See also

Created on May 3, 2022 at 21:34:10. See the history of this page for a list of all contributions to it.