Theorems · Definition · category theory

CategoryTheory.Limits.equalizer.lift

{C : Type u} →
  {X Y : C} →
    [inst : CategoryTheory.Category.{v, u} C] →
      {f g : X ⟶ Y} →
        [inst_1 : CategoryTheory.Limits.HasEqualizer f g] →
          {W : C} →
            (k : W ⟶ X) →
              CategoryTheory.CategoryStruct.comp k f = CategoryTheory.CategoryStruct.comp k g →
                (W ⟶ CategoryTheory.Limits.equalizer f g)

A morphism k : W ⟶ X satisfying k ≫ f = k ≫ g factors through the equalizer of f and g via equalizer.lift : W ⟶ equalizer f g.

Defined in: Mathlib.CategoryTheory.Limits.Shapes.Equalizers
Cited by: 19 results in Mathlib
Foundations: Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasEqualizer

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites8

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.parallelPairproof · cited by 766
CategoryTheory.Limits.Fork.ofιproof · cited by 66
CategoryTheory.Limits.equalizerstatement · cited by 60
CategoryTheory.Limits.limit.liftproof · cited by 48
CategoryTheory.Limits.HasEqualizerstatement and proof · cited by 47

Cited by23

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.equalizerPullbackMapIsoproof · cited by 10
CategoryTheory.Limits.equalizerComparisonproof · cited by 6
CategoryTheory.Limits.image.extproof · cited by 4
CategoryTheory.Limits.equalizer.lift_ιstatement · cited by 4
CategoryTheory.Comonad.ComonadicityInternal.comparisonRightAdjointHomEquivproof · cited by 4
AlgebraicGeometry.ext_of_isDominant_of_isSeparatedproof · cited by 3
CategoryTheory.Limits.equalizerSubobject_factorsproof · cited by 2
CategoryTheory.Limits.equalizerPullbackMapIso_hom_fstproof · cited by 1
CategoryTheory.Limits.equalizerPullbackMapIso_hom_sndproof · cited by 1
AlgebraicGeometry.ext_of_fromSpecResidueField_eqproof · cited by 1
CategoryTheory.Limits.equalizer.isoSourceOfSelf_invstatement · cited by 1
CategoryTheory.Limits.map_lift_equalizerComparisonstatement and proof · cited by 1