Theorems · Definition · category theory

CategoryTheory.Limits.diagramIsoPair

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (F : CategoryTheory.Functor (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) C) →
      F ≅
        CategoryTheory.Limits.pair (F.obj { as := CategoryTheory.Limits.WalkingPair.left })
          (F.obj { as := CategoryTheory.Limits.WalkingPair.right })

Every functor out of the walking pair is naturally isomorphic (actually, equal) to a pair

Defined in: Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
Cited by: 17 results in Mathlib
Foundations: Depth 25 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.

Cites9

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
CategoryTheory.Functor.objstatement and proof · cited by 19,642
CategoryTheory.Functorstatement and proof · cited by 16,252
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.Discretestatement and proof · cited by 2,447
CategoryTheory.Limits.WalkingPairstatement and proof · cited by 1,319
CategoryTheory.Iso.reflproof · cited by 727
CategoryTheory.Limits.pairstatement · cited by 536
CategoryTheory.Limits.mapPairIsoproof · cited by 2

Cited by22

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.pairCompproof · cited by 6
CategoryTheory.Limits.preservesBinaryBiproduct_of_preservesBinaryProductproof · cited by 2
CategoryTheory.Limits.isLimitMapConeBinaryFanEquivproof · cited by 1
CategoryTheory.Limits.preservesBinaryBiproduct_of_preservesBinaryCoproductproof · cited by 1
CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquivproof · cited by 1
CategoryTheory.preservesBinaryProducts_of_exponentialIdealproof · cited by 1
CategoryTheory.Limits.diagramIsoPair_hom_appstatement and proof · cited by 1
CategoryTheory.Limits.preservesBinaryBiproduct_of_preservesBiproductproof · cited by 1
CategoryTheory.Limits.preservesBinaryCoproducts_of_isIso_coprodComparisonproof · cited by 1
CategoryTheory.Limits.preservesBinaryCoproducts_of_preservesBinaryBiproductsproof · cited by 0
CategoryTheory.Limits.preservesBinaryProduct_of_preservesBinaryBiproductproof · cited by 0
CategoryTheory.Limits.preservesBinaryProducts_of_isIso_prodComparisonproof · cited by 0