Theorems · Definition · ring theory

BialgEquiv.ofBialgHom

{R : Type u} →
  {A : Type v} →
    {B : Type w} →
      [inst : CommSemiring R] →
        [inst_1 : Semiring A] →
          [inst_2 : Semiring B] →
            [inst_3 : Algebra R A] →
              [inst_4 : Algebra R B] →
                [inst_5 : CoalgebraStruct R A] →
                  [inst_6 : CoalgebraStruct R B] →
                    (f : A →ₐc[R] B) →
                      (g : B →ₐc[R] A) → f.comp g = BialgHom.id R B → g.comp f = BialgHom.id R A → A ≃ₐc[R] B

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

Defined in

Mathlib.RingTheory.Bialgebra.Equiv

Cited by

2 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringSemiringAlgebraAlgebraCoalgebraStructCoalgebraStruct