Theorems · Definition · category theory

CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.mkMonoidalCategoryStruct

(C : Type u₁) →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    (V : Type u₂) →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} V] →
        [inst_2 : CategoryTheory.MonoidalCategory C] →
          [inst_3 : CategoryTheory.MonoidalCategory V] →
            (D : Type u₃) →
              [inst_4 : CategoryTheory.Category.{v₃, u₃} D] →
                [CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D] →
                  [∀ (v : V) (d : C),
                        CategoryTheory.Limits.PreservesColimitsOfShape
                          (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
                          (CategoryTheory.MonoidalCategory.tensorLeft v)] →
                    [∀ (v : V) (d : C),
                          CategoryTheory.Limits.PreservesColimitsOfShape
                            (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
                            (CategoryTheory.MonoidalCategory.tensorRight v)] →
                      [∀ (v : V) (d : C),
                            CategoryTheory.Limits.PreservesColimitsOfShape
                              (CategoryTheory.CostructuredArrow
                                (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
                                d)
                              (CategoryTheory.MonoidalCategory.tensorRight v)] →
                        [∀ (v : V) (d : C),
                              CategoryTheory.Limits.PreservesColimitsOfShape
                                (CategoryTheory.CostructuredArrow
                                  (CategoryTheory.Functor.fromPUnit
                                    (CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
                                  d)
                                (CategoryTheory.MonoidalCategory.tensorLeft v)] →
                          CategoryTheory.MonoidalCategoryStruct D

Given a fully faithful functor ι : C ⥤ V ⥤ D, a family of Day convolutions, candidate functions for tensorObj and tensorHom, suitable isomorphisms ι.obj (tensorObj d d') ≅ ι.obj (tensorObj d) ⊛ ι.obj (tensorObj d') that behave in a lawful way with respect to the chosen Day convolutions, we can construct a MonoidalCategoryStruct on D.

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution

Cited by

3 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MonoidalCategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCoreCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShape