Theorems · Definition · category theory

HomologicalComplex.mapBifunctor

{C₁ : Type u_1} →
  {C₂ : Type u_2} →
    {D : Type u_3} →
      [inst : CategoryTheory.Category.{v_1, u_1} C₁] →
        [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] →
          [inst_2 : CategoryTheory.Category.{v_3, u_3} D] →
            {I₁ : Type u_4} →
              {I₂ : Type u_5} →
                {J : Type u_6} →
                  {c₁ : ComplexShape I₁} →
                    {c₂ : ComplexShape I₂} →
                      [inst_3 : CategoryTheory.Limits.HasZeroMorphisms C₁] →
                        [inst_4 : CategoryTheory.Limits.HasZeroMorphisms C₂] →
                          [inst_5 : CategoryTheory.Preadditive D] →
                            (K₁ : HomologicalComplex C₁ c₁) →
                              (K₂ : HomologicalComplex C₂ c₂) →
                                (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ D)) →
                                  [inst_6 : F.PreservesZeroMorphisms] →
                                    [inst_7 : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms] →
                                      (c : ComplexShape J) →
                                        [inst_8 : TotalComplexShape c₁ c₂ c] →
                                          [K₁.HasMapBifunctor K₂ F c] → [DecidableEq J] → HomologicalComplex D c

Given K₁ : HomologicalComplex C₁ c₁, K₂ : HomologicalComplex C₂ c₂, a bifunctor F : C₁ ⥤ C₂ ⥤ D and a complex shape ComplexShape J such that we have [TotalComplexShape c₁ c₂ c], this mapBifunctor K₁ K₂ F c : HomologicalComplex D c is the total complex of the bicomplex obtained by applying F to K₁ and K₂.

Defined in

Mathlib.Algebra.Homology.Bifunctor

Cited by

109 results in Mathlib

Foundations

Depth 45 from the axioms, rests on 772 definitions · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.PreadditiveCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsTotalComplexShapeHomologicalComplex.HasMapBifunctorDecidableEq