Theorems · Definition · category theory

CategoryTheory.Mat_.additiveObjIsoBiproduct

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {D : Type u₁} →
        [inst_2 : CategoryTheory.Category.{v₁, u₁} D] →
          [inst_3 : CategoryTheory.Preadditive D] →
            (F : CategoryTheory.Functor (CategoryTheory.Mat_ C) D) →
              [inst_4 : F.Additive] →
                (M : CategoryTheory.Mat_ C) → F.obj M ≅ ⨁ fun i => F.obj ((CategoryTheory.Mat_.embedding C).obj (M.X i))

Every M is a direct sum of objects from C, and F preserves biproducts.

Defined in

Mathlib.CategoryTheory.Preadditive.Mat

Cited by

8 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Functor.Additive