Theorems · Definition · category theory

CategoryTheory.MonoidalCategory.DayFunctor.isoPointwiseLeftKanExtension

{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] →
            [hasDayConvolution :
                ∀ (F G : CategoryTheory.Functor C V),
                  (CategoryTheory.MonoidalCategory.tensor C).HasPointwiseLeftKanExtension
                    (CategoryTheory.MonoidalCategory.externalProduct F G)] →
              [hasDayConvolutionUnit :
                  (CategoryTheory.Functor.fromPUnit
                        (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).HasPointwiseLeftKanExtension
                    (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit V))] →
                [inst_4 :
                    ∀ (v : V) (d : C),
                      CategoryTheory.Limits.PreservesColimitsOfShape
                        (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
                        (CategoryTheory.MonoidalCategory.tensorLeft v)] →
                  [inst_5 :
                      ∀ (v : V) (d : C),
                        CategoryTheory.Limits.PreservesColimitsOfShape
                          (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
                          (CategoryTheory.MonoidalCategory.tensorRight v)] →
                    [inst_6 :
                        ∀ (v : V) (d : C),
                          CategoryTheory.Limits.PreservesColimitsOfShape
                            (CategoryTheory.CostructuredArrow
                              (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) d)
                            (CategoryTheory.MonoidalCategory.tensorLeft v)] →
                      [inst_7 :
                          ∀ (v : V) (d : C),
                            CategoryTheory.Limits.PreservesColimitsOfShape
                              (CategoryTheory.CostructuredArrow
                                (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
                                d)
                              (CategoryTheory.MonoidalCategory.tensorRight v)] →
                        [inst_8 :
                            ∀ (v : V) (d : C × C),
                              CategoryTheory.Limits.PreservesColimitsOfShape
                                (CategoryTheory.CostructuredArrow
                                  ((CategoryTheory.Functor.id C).prod
                                    (CategoryTheory.Functor.fromPUnit
                                      (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)))
                                  d)
                                (CategoryTheory.MonoidalCategory.tensorRight v)] →
                          [inst_9 :
                              ∀ (v : V) (d : C × C),
                                CategoryTheory.Limits.PreservesColimitsOfShape
                                  (CategoryTheory.CostructuredArrow
                                    ((CategoryTheory.MonoidalCategory.tensor C).prod (CategoryTheory.Functor.id C)) d)
                                  (CategoryTheory.MonoidalCategory.tensorRight v)] →
                            (F G : CategoryTheory.MonoidalCategory.DayFunctor C V) →
                              (CategoryTheory.MonoidalCategoryStruct.tensorObj F G).functor ≅
                                (CategoryTheory.MonoidalCategory.tensor C).pointwiseLeftKanExtension
                                  (CategoryTheory.MonoidalCategory.externalProduct F.functor G.functor)

An abstract isomorphism between (F ⊗ G).functor and the generic pointwise left Kan extension of F.functor ⊠ G.functor along the

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor

Cited by

2 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MonoidalCategoryCategoryTheory.Functor.HasPointwiseLeftKanExtensionCategoryTheory.Functor.HasPointwiseLeftKanExtensionCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShape