Theorems · Theorem · category theory

CategoryTheory.ShortComplex.Exact.shortExact

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  {S : CategoryTheory.ShortComplex C} (hS : S.Exact) (h : S.HomologyData),
  { X₁ := h.left.K, X₂ := S.X₂, X₃ := h.right.Q, f := h.left.i, g := h.right.p, zero := ⋯ }.ShortExact

Is S is an exact short complex and h : S.HomologyData, there is a short exact sequence 0 ⟶ h.left.K ⟶ S.X₂ ⟶ h.right.Q ⟶ 0.

Defined in: Mathlib.Algebra.Homology.ShortComplex.ShortExact
Cited by: 1 results in Mathlib
Foundations: Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites33

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
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement · cited by 17,999
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.Preadditivestatement and proof · cited by 3,309
CategoryTheory.Category.comp_idproof · cited by 2,119
CategoryTheory.Category.id_compproof · cited by 1,998
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
CategoryTheory.Monoproof · cited by 893
CategoryTheory.ShortComplex.X₁proof · cited by 889
CategoryTheory.ShortComplex.X₃proof · cited by 876

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.ObjectProperty.prop_X₂_of_exactproof · cited by 0