Theorems · Theorem · category theory

TopCat.Sheaf.sections_exact_of_left_exact

∀ {X : TopCat} {U : TopologicalSpace.Opens ↑X} {S : CategoryTheory.ShortComplex (TopCat.Sheaf AddCommGrpCat X)},
  S.Exact →
    CategoryTheory.Mono S.f →
      ∀ (s : ↑(S.X₂.obj.obj (Opposite.op U))),
        (CategoryTheory.ConcreteCategory.hom (S.g.hom.app (Opposite.op U))) s = 0 →
          ∃ t, (CategoryTheory.ConcreteCategory.hom (S.f.hom.app (Opposite.op U))) t = s

Defined in: Mathlib.Topology.Sheaves.AddCommGrpCat
Cited by: 1 results in Mathlib
Foundations: Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement · cited by 53,352
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Oppositestatement · cited by 8,081
CategoryTheory.NatTrans.appstatement and proof · cited by 7,406
CategoryTheory.Functor.compstatement · cited by 6,529
CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
AddMonoidHomstatement · cited by 3,230
TopCat.carrierstatement and proof · cited by 3,184
TopologicalSpace.Opensstatement and proof · cited by 2,040
TopCatstatement and proof · cited by 1,889

Cited by1

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TopCat.Sheaf.IsFlasque.epi_of_shortExactproof · cited by 1