Theorems · Definition · algebraic geometry
AlgebraicGeometry.Scheme.Pullback.gluing
{X Y Z : AlgebraicGeometry.Scheme} →
(𝒰 : X.OpenCover) →
(f : X ⟶ Z) →
(g : Y ⟶ Z) →
[∀ (i : 𝒰.I₀), CategoryTheory.Limits.HasPullback (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g] →
AlgebraicGeometry.Scheme.GlueDataGiven Uᵢ ×[Z] Y, this is the glued fibred product X ×[Z] Y.
- Defined in
- Mathlib.AlgebraicGeometry.Pullbacks
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 147 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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.pullbackproof · 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 by37
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.Pullback.p1statement and proof · cited by 18
- AlgebraicGeometry.Scheme.Pullback.pullbackP1Isostatement and proof · cited by 10
- AlgebraicGeometry.Scheme.Pullback.p2statement and proof · cited by 7
- AlgebraicGeometry.Scheme.Pullback.pullbackFstιToVstatement and proof · cited by 5
- AlgebraicGeometry.Scheme.Pullback.gluedLiftPullbackMapstatement · cited by 4
- AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_inv_fststatement and proof · cited by 2
- AlgebraicGeometry.Scheme.Pullback.gluedLiftstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.Pullback.pullbackFstιToV_fststatement and proof · cited by 2
- AlgebraicGeometry.Scheme.Pullback.pullbackFstιToV_sndstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_fststatement · cited by 1
- AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_sndstatement · cited by 1
- AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_ιstatement and proof · cited by 1