Theorems · Definition · algebraic geometry

AlgebraicGeometry.Scheme.Pullback.fV

{X Y Z : AlgebraicGeometry.Scheme} →
  (𝒰 : X.OpenCover) →
    (f : X ⟶ Z) →
      (g : Y ⟶ Z) →
        [inst : ∀ (i : 𝒰.I₀), CategoryTheory.Limits.HasPullback (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g] →
          (i j : 𝒰.I₀) →
            AlgebraicGeometry.Scheme.Pullback.v 𝒰 f g i j ⟶
              CategoryTheory.Limits.pullback (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g

The inclusion map of V i j = (Uᵢ ×[Z] Y) ×[X] Uⱼ ⟶ Uᵢ ×[Z] Y

Defined in: Mathlib.AlgebraicGeometry.Pullbacks
Cited by: 21 results in Mathlib
Foundations: Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Limits.HasPullback

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites14

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

Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CategoryTheory.Limits.pullbackstatement · cited by 864
CategoryTheory.PreZeroHypercover.I₀statement and proof · cited by 763
CategoryTheory.PreZeroHypercover.Xstatement · cited by 649
CategoryTheory.Limits.pullback.fstproof · cited by 639
CategoryTheory.PreZeroHypercover.fstatement and proof · cited by 542
AlgebraicGeometry.IsOpenImmersionstatement · cited by 476
CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercoverstatement and proof · cited by 469
CategoryTheory.Limits.HasPullbackstatement and proof · cited by 434
AlgebraicGeometry.Scheme.precoveragestatement · cited by 336

Cited by22

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.Pullback.t'statement and proof · cited by 20
AlgebraicGeometry.Scheme.Pullback.t'_fst_sndstatement and proof · cited by 5
AlgebraicGeometry.Scheme.Pullback.t'_snd_sndstatement and proof · cited by 5
AlgebraicGeometry.Scheme.Pullback.t'_fst_fst_fststatement and proof · cited by 4
AlgebraicGeometry.Scheme.Pullback.t'_fst_fst_sndstatement and proof · cited by 2
AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_fststatement and proof · cited by 2
AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_sndstatement and proof · cited by 2
AlgebraicGeometry.Scheme.Pullback.cocycle_fst_fst_fststatement and proof · cited by 1
AlgebraicGeometry.Scheme.Pullback.cocycle_fst_fst_sndstatement and proof · cited by 1
AlgebraicGeometry.Scheme.Pullback.cocycle_fst_sndstatement and proof · cited by 1
AlgebraicGeometry.Scheme.Pullback.cocycle_snd_fst_fststatement and proof · cited by 1
AlgebraicGeometry.Scheme.Pullback.cocycle_snd_fst_sndstatement and proof · cited by 1