Theorems · Theorem · category theory

CategoryTheory.HomOrthogonal.matrixDecomposition_symm_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 : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (CategoryTheory.End (s i))),
  o.matrixDecomposition.symm z =
    CategoryTheory.Limits.biproduct.matrix fun j k =>
      if h : f j = g k then CategoryTheory.CategoryStruct.comp (z (f j) ⟨k, ⋯⟩ ⟨j, ⋯⟩) (CategoryTheory.eqToHom ⋯) else 0

Defined in

Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasFiniteBiproductsFiniteFinite