Theorems · Definition · category theory

CommAlgCat.isoEquivAlgEquiv

{R : Type u} →
  [inst : CommRing R] →
    {X Y : Type v} →
      [inst_1 : CommRing X] →
        [inst_2 : Algebra R X] →
          [inst_3 : CommRing Y] → [inst_4 : Algebra R Y] → (CommAlgCat.of R X ≅ CommAlgCat.of R Y) ≃ X ≃ₐ[R] Y

Algebra equivalences between Algebras are the same as isomorphisms in CommAlgCat.

Defined in: Mathlib.Algebra.Category.CommAlgCat.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 33 from the axioms · uses propext, Quot.sound
Assumes: CommRing CommRing Algebra CommRing Algebra

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
Equivstatement · cited by 8,337
CategoryTheory.Isostatement · cited by 3,963
AlgEquivstatement · cited by 1,681
CommAlgCatstatement · cited by 96
CommAlgCat.ofstatement · cited by 33
CommAlgCat.isoMkproof · cited by 4
CommAlgCat.algEquivOfIsoproof · cited by 3

Cited by2

Results whose statement or proof uses this declaration.

CommAlgCat.isoEquivAlgEquiv_applystatement and proof · cited by 0
CommAlgCat.isoEquivAlgEquiv_symm_applystatement and proof · cited by 0