Theorems · Definition · category theory

CategoryTheory.ShortComplex.Exact.fIsKernel

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {S : CategoryTheory.ShortComplex C} →
        [CategoryTheory.Balanced C] →
          S.Exact →
            [CategoryTheory.Mono S.f] → CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι S.f ⋯)

In a balanced category, if a short complex S is exact and S.f is a mono, then S.X₁ is the kernel of S.g.

Defined in

Mathlib.Algebra.Homology.ShortComplex.Exact

Cited by

12 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.BalancedCategoryTheory.Mono