Theorems · Theorem · category theory

CategoryTheory.HomOrthogonal.equiv_of_iso

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {ι : Type u_1} {s : ι → C}
  [inst_1 : CategoryTheory.Preadditive C] [inst_2 : CategoryTheory.Limits.HasFiniteBiproducts C]
  [∀ (i : ι), InvariantBasisNumber (CategoryTheory.End (s i))],
  CategoryTheory.HomOrthogonal s →
    ∀ {α β : Type} [inst_4 : Finite α] [inst_5 : Finite β] {f : α → ι} {g : β → ι}
      (i : (⨁ fun a => s (f a)) ≅ ⨁ fun b => s (g b)), ∃ e, ∀ (a : α), g (e a) = f a

Given a hom orthogonal family s : ι → C for which each End (s i) is a ring with invariant basis number (e.g. if each s i is simple), if two direct sums over s are isomorphic, then they have the same multiplicities.

Defined in

Mathlib.CategoryTheory.Preadditive.HomOrthogonal

Cited by

0 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasFiniteBiproductsInvariantBasisNumberFiniteFinite