Theorems · Definition · category theory

CategoryTheory.CartesianMonoidalCategory.tensorProductIsBinaryProduct

{C : Type u} →
  {inst : CategoryTheory.Category.{v, u} C} →
    [self : CategoryTheory.CartesianMonoidalCategory C] →
      (X Y : C) →
        CategoryTheory.Limits.IsLimit
          (CategoryTheory.Limits.BinaryFan.mk (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)
            (CategoryTheory.SemiCartesianMonoidalCategory.snd X Y))

The monoidal product is the categorical product.

Defined in: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
Cited by: 6 results in Mathlib
Foundations: Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.CartesianMonoidalCategory

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites10

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.MonoidalCategoryStruct.tensorObjstatement · cited by 3,106
CategoryTheory.Discretestatement · cited by 2,447
CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
CategoryTheory.CartesianMonoidalCategorystatement and proof · cited by 947
CategoryTheory.Limits.IsLimitstatement · cited by 664
CategoryTheory.Limits.pairstatement · cited by 536
CategoryTheory.SemiCartesianMonoidalCategory.fststatement · cited by 184
CategoryTheory.SemiCartesianMonoidalCategory.sndstatement · cited by 181
CategoryTheory.Limits.BinaryFan.mkstatement · cited by 112

Cited by12

Results whose statement or proof uses this declaration.

CategoryTheory.CartesianMonoidalCategory.liftproof · cited by 160
CategoryTheory.CartesianMonoidalCategory.hom_extproof · cited by 46
CategoryTheory.CartesianMonoidalCategory.lift_sndproof · cited by 37
CategoryTheory.CartesianMonoidalCategory.lift_fstproof · cited by 36
CategoryTheory.CartesianMonoidalCategory.prodComparisonIsoproof · cited by 16
CategoryTheory.CartesianMonoidalCategory.tensorLeftIsoProdproof · cited by 0
CategoryTheory.Sheaf.tensorProd_isSheafproof · cited by 0
CategoryTheory.CartesianMonoidalCategory.fullSubcategory_tensorProductIsBinaryProduct_lift_homstatement and proof · cited by 0
CategoryTheory.Functor.chosenProd.isLimitproof · cited by 0
CategoryTheory.CartesianMonoidalCategory.ofReflectiveproof · cited by 0
CategoryTheory.CartesianMonoidalCategory.preservesLimit_pair_of_isIso_prodComparisonproof · cited by 0
CategoryTheory.CartesianMonoidalCategory.isLimitCartesianMonoidalCategoryOfPreservesLimitsproof · cited by 0