Theorems · Definition · category theory

CategoryTheory.ShortComplex.homologyIsoImageICyclesCompPOpcycles

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      (S : CategoryTheory.ShortComplex C) →
        S.homology ≅ CategoryTheory.Limits.image (CategoryTheory.CategoryStruct.comp S.iCycles S.pOpcycles)

The homology of a short complex S in an abelian category identifies to the image of S.iCycles ≫ S.pOpcycles : S.cycles ⟶ S.opcycles.

Defined in

Mathlib.Algebra.Homology.ShortComplex.Abelian

Cited by

3 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian