Theorems · Definition · category theory

CategoryTheory.Equivalence.IsMonoidal

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    [inst_1 : CategoryTheory.MonoidalCategory C] →
      {D : Type u₂} →
        [inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
          [inst_3 : CategoryTheory.MonoidalCategory D] →
            (e : C ≌ D) → [e.functor.Monoidal] → [e.inverse.Monoidal] → Prop

An equivalence of categories involving monoidal functors is monoidal if the underlying adjunction satisfies certain compatibilities with respect to the monoidal functor data.

Defined in: Mathlib.CategoryTheory.Monoidal.Functor
Cited by: 40 results in Mathlib
Foundations: Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.MonoidalCategory CategoryTheory.Category CategoryTheory.MonoidalCategory CategoryTheory.Functor.Monoidal CategoryTheory.Functor.Monoidal

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites8

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

CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.MonoidalCategorystatement and proof · cited by 3,095
CategoryTheory.Equivalence.functorstatement and proof · cited by 1,268
CategoryTheory.Equivalence.inversestatement and proof · cited by 1,130
CategoryTheory.Equivalencestatement and proof · cited by 601
CategoryTheory.Functor.Monoidalstatement and proof · cited by 288
CategoryTheory.Equivalence.toAdjunctionproof · cited by 60
CategoryTheory.Adjunction.IsMonoidalproof · cited by 20

Cited by43

Results whose statement or proof uses this declaration.

CategoryTheory.Equivalence.mapAddMonstatement and proof · cited by 4
CategoryTheory.Equivalence.mapCommMonstatement and proof · cited by 4
CategoryTheory.Equivalence.mapMonstatement and proof · cited by 4
CategoryTheory.Equivalence.functor_map_ε_inverse_comp_counitIso_hom_appstatement and proof · cited by 3
CategoryTheory.Equivalence.counitIso_inv_app_tensor_comp_functor_map_δ_inversestatement and proof · cited by 2
CategoryTheory.Equivalence.counitInv_app_comp_functor_map_η_inversestatement and proof · cited by 1
CategoryTheory.Equivalence.counitInv_app_tensor_comp_functor_map_δ_inversestatement and proof · cited by 1
CategoryTheory.Equivalence.counitIso_inv_app_comp_functor_map_η_inversestatement and proof · cited by 1
CategoryTheory.Equivalence.functor_map_ε_inverse_comp_counit_appstatement and proof · cited by 1
CategoryTheory.Equivalence.functor_map_μ_inverse_comp_counitIso_hom_app_tensorstatement and proof · cited by 1
CategoryTheory.Equivalence.functor_map_μ_inverse_comp_counit_app_tensorstatement and proof · cited by 1
CategoryTheory.Equivalence.unitIso_hom_app_comp_inverse_map_η_functorstatement and proof · cited by 1