Theorems · Definition · category theory

CategoryTheory.HomOrthogonal.matrixDecomposition

{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] →
            CategoryTheory.HomOrthogonal s →
              {α β : Type} →
                [inst_3 : Finite α] →
                  [inst_4 : Finite β] →
                    {f : α → ι} →
                      {g : β → ι} →
                        ((⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)) ≃
                          ((i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (CategoryTheory.End (s i)))

Morphisms between two direct sums over a hom orthogonal family s : ι → C are equivalent to block diagonal matrices, with blocks indexed by ι, and matrix entries in i-th block living in the endomorphisms of s i.

Defined in

Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Cited by

6 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasFiniteBiproductsFiniteFinite