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Theorems · Theorem · category theory

ModuleCat.shortComplex_shortExact

∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)),
  Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom S.f) ⇑(CategoryTheory.ConcreteCategory.hom S.g) →
    Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom S.f) →
      Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom S.g) → S.ShortExact
Defined in
Mathlib.Algebra.Homology.ShortComplex.ModuleCat
Cited by
1 results in Mathlib
Foundations
Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
Ring
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Around this declaration

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ModuleCat.shortComplexOfConj_shortExact · cited by 0ModuleCat.shortComplexOfC…DFunLike.coe · cited by 62936DFunLike.coeRingHom.id · cited by 18349RingHom.idLinearMap · cited by 10215LinearMapRing · cited by 7463RingCategoryTheory.ConcreteCategory.hom · cited by 4022ConcreteCategory.homCategoryTheory.ShortComplex · cited by 1850CategoryTheory.ShortCompl…ModuleCat · cited by 1429ModuleCatCategoryTheory.ShortComplex.X₂ · cited by 1115ShortComplex.X₂ModuleCat.carrier · cited by 997ModuleCat.carrierCategoryTheory.ShortComplex.X₁ · cited by 889ShortComplex.X₁CategoryTheory.ShortComplex.X₃ · cited by 876ShortComplex.X₃CategoryTheory.ShortComplex.g · cited by 658ShortComplex.gCategoryTheory.ShortComplex.f · cited by 653ShortComplex.fCategoryTheory.ShortComplex.ShortExact · cited by 232ShortComplex.ShortExactFunction.Exact · cited by 182Function.ExactModuleCat.shortComplex_shortE…CITED BYCITES

Cites16

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