Theorems · Definition · ring theory

CoalgEquiv.ofCoalgHom

{R : Type u_1} →
  {A : Type u_2} →
    {B : Type u_3} →
      [inst : CommSemiring R] →
        [inst_1 : AddCommMonoid A] →
          [inst_2 : AddCommMonoid B] →
            [inst_3 : Module R A] →
              [inst_4 : Module R B] →
                [inst_5 : CoalgebraStruct R A] →
                  [inst_6 : CoalgebraStruct R B] →
                    (f : A →ₗc[R] B) →
                      (g : B →ₗc[R] A) → f.comp g = CoalgHom.id R B → g.comp f = CoalgHom.id R A → A ≃ₗc[R] B

If a coalgebra morphism has an inverse, it is a coalgebra isomorphism.

Defined in

Mathlib.RingTheory.Coalgebra.Equiv

Cited by

2 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleCoalgebraStructCoalgebraStruct