Theorems · Theorem · category theory

CategoryTheory.HomOrthogonal.matrixDecomposition_apply

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {ι : Type u_1} {s : ι → C}
  [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_2 : CategoryTheory.Limits.HasFiniteBiproducts C]
  (o : CategoryTheory.HomOrthogonal s) {α β : Type} [inst_3 : Finite α] [inst_4 : Finite β] {f : α → ι} {g : β → ι}
  (z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)) (i : ι) (j : ↑(g ⁻¹' {i})) (k : ↑(f ⁻¹' {i})),
  o.matrixDecomposition z i j k =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.components z ↑k ↑j)
        (CategoryTheory.eqToHom ⋯))

Defined in

Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Cited by

2 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasFiniteBiproductsFiniteFinite