Theorems · Definition · category theory
CategoryTheory.Mat_.isoBiproductEmbedding
{C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.Preadditive C] →
(M : CategoryTheory.Mat_ C) → M ≅ ⨁ fun i => (CategoryTheory.Mat_.embedding C).obj (M.X i)Every object in Mat_ C is isomorphic to the biproduct of its summands.
- Defined in
- Mathlib.CategoryTheory.Preadditive.Mat
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- CategoryTheory.Functor.objstatement and proof · cited by 19,642
- CategoryTheory.Isostatement · cited by 3,963
- CategoryTheory.Preadditivestatement and proof · cited by 3,309
- CategoryTheory.eqToHomproof · cited by 860
- CategoryTheory.Limits.biproductstatement · cited by 188
- CategoryTheory.Mat_statement and proof · cited by 44
- CategoryTheory.Limits.biproduct.liftproof · cited by 31
- CategoryTheory.Mat_.ιstatement and proof · cited by 29
- CategoryTheory.Limits.biproduct.descproof · cited by 29
- CategoryTheory.Mat_.Xstatement and proof · cited by 24
- CategoryTheory.Mat_.embeddingstatement and proof · cited by 17
Cited by7
Results whose statement or proof uses this declaration.
- CategoryTheory.Mat_.additiveObjIsoBiproductproof · cited by 8
- CategoryTheory.Mat_.additiveObjIsoBiproduct_hom_πstatement · cited by 2
- CategoryTheory.Mat_.isoBiproductEmbedding_homstatement and proof · cited by 1
- CategoryTheory.Mat_.isoBiproductEmbedding_invstatement and proof · cited by 1
- CategoryTheory.Mat_.ι_additiveObjIsoBiproduct_invstatement · cited by 1
- CategoryTheory.Mat_.ι_additiveObjIsoBiproduct_inv_assocstatement and proof · cited by 1
- CategoryTheory.Mat_.additiveObjIsoBiproduct_hom_π_assocstatement and proof · cited by 0