Theorems · Theorem · category theory

TopCat.Presheaf.sections_exact_of_exact

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

Defined in: Mathlib.Topology.Sheaves.AddCommGrpCat
Cited by: 1 results in Mathlib
Foundations: Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites25

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
CategoryTheory.Functor.objstatement and proof · cited by 19,642
CategoryTheory.Functorproof · cited by 16,252
Oppositestatement and proof · cited by 8,081
CategoryTheory.NatTrans.appstatement and proof · cited by 7,406
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
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115

Cited by1

Results whose statement or proof uses this declaration.

TopCat.Sheaf.sections_exact_of_left_exactproof · cited by 1