Library CT.Instance.Functor.FaithfulFunctor

Require Import CT.Category.
Require Import CT.Functor.

Faithful Functors

A "faithful" functor is one that is injective on hom-sets. That is, A functor

$F : C \to D$ is faithful if

$\forall x, y \in C$ , the function

$F : C(x, y) \to D(F(x), F(y))$ is injective.

So we require a Functor and a proof that injectivity of hom-sets is satisfied.

Section FaithfulFunctor.
  Context {A B : Category}.
  Variable (F : Functor A B).

  Definition FaithfulFunctor :=
    ∀ {a b} (f : mor A a b) (g : mor A a b),
      F_mor F f = F_mor F g → f = g.
End FaithfulFunctor.