Theorems · Definition · category theory

CategoryTheory.Limits.FormalCoproduct.pullbackCone

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {X Y Z : CategoryTheory.Limits.FormalCoproduct C} →
      (f : X ⟶ Z) →
        (g : Y ⟶ Z) →
          ((i : Function.Pullback f.f g.f) →
              CategoryTheory.Limits.PullbackCone
                (CategoryTheory.CategoryStruct.comp (f.φ (↑i).1) (CategoryTheory.eqToHom ⋯)) (g.φ (↑i).2)) →
            CategoryTheory.Limits.PullbackCone f g

Given two morphisms f : X ⟶ Z and g : Y ⟶ Z, given pullback in C over each component, construct the pullback in FormalCategory.{w} C.

Defined in: Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic
Cited by: 10 results in Mathlib
Foundations: Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category

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

Cites15

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

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Limits.Cone.ptproof · cited by 1,298
CategoryTheory.eqToHomstatement and proof · cited by 860
CategoryTheory.Limits.PullbackCone.mkproof · cited by 203
CategoryTheory.Limits.PullbackConestatement and proof · cited by 136
CategoryTheory.Limits.FormalCoproductstatement and proof · cited by 122
CategoryTheory.Limits.PullbackCone.fstproof · cited by 118
CategoryTheory.Limits.PullbackCone.sndproof · cited by 113
CategoryTheory.Limits.FormalCoproduct.Istatement and proof · cited by 87
CategoryTheory.Limits.FormalCoproduct.objstatement · cited by 72

Cited by12

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.FormalCoproduct.homPullbackEquivstatement and proof · cited by 3
CategoryTheory.Limits.FormalCoproduct.isLimitPullbackConestatement · cited by 2
CategoryTheory.Limits.FormalCoproduct.pullbackCone_conditionstatement and proof · cited by 1
CategoryTheory.Limits.FormalCoproduct.hasPullback_of_pullbackConeproof · cited by 0
CategoryTheory.Limits.FormalCoproduct.homPullbackEquiv_apply_coestatement and proof · cited by 0
CategoryTheory.Limits.FormalCoproduct.homPullbackEquiv_symm_apply_f_coestatement · cited by 0
CategoryTheory.Limits.FormalCoproduct.homPullbackEquiv_symm_apply_φstatement · cited by 0
CategoryTheory.Limits.FormalCoproduct.pullbackCone_fst_fstatement and proof · cited by 0
CategoryTheory.Limits.FormalCoproduct.pullbackCone_fst_φstatement and proof · cited by 0
CategoryTheory.Limits.FormalCoproduct.pullbackCone_snd_fstatement and proof · cited by 0
CategoryTheory.Limits.FormalCoproduct.pullbackCone_snd_φstatement and proof · cited by 0
CategoryTheory.Limits.FormalCoproduct.isPullbackstatement · cited by 0