Theorems · Definition · category theory

CategoryTheory.Biprod.gaussian

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] →
        {X₁ X₂ Y₁ Y₂ : C} →
          (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) →
            [CategoryTheory.IsIso
                  (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl
                    (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.fst))] →
              (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) ×'
                (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) ×'
                  (g₂₂ : X₂ ⟶ Y₂) ×'
                    CategoryTheory.CategoryStruct.comp L.hom (CategoryTheory.CategoryStruct.comp f R.hom) =
                      CategoryTheory.Limits.biprod.map
                        (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl
                          (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.fst))
                        g₂₂

If f is a morphism X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂ whose X₁ ⟶ Y₁ entry is an isomorphism, then we can construct isomorphisms L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂ and R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂ so that L.hom ≫ g ≫ R.hom is diagonal (with X₁ ⟶ Y₁ component still f), via Gaussian elimination.

Defined in

Mathlib.CategoryTheory.Preadditive.Biproducts

Cited by

0 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproductsCategoryTheory.IsIso