Theorems · Definition · category theory

CategoryTheory.zeroMul

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.CartesianMonoidalCategory C] →
      {A : C} →
        [CategoryTheory.Closed A] →
          {I : C} → CategoryTheory.Limits.IsInitial I → (CategoryTheory.MonoidalCategoryStruct.tensorObj A I ≅ I)

If an initial object I exists in a CCC, then A ⨯ I ≅ I.

Defined in: Mathlib.CategoryTheory.Monoidal.Closed.Cartesian
Cited by: 12 results in Mathlib
Foundations: Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.CartesianMonoidalCategory CategoryTheory.Closed

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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
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.MonoidalCategoryStruct.tensorObjstatement and proof · cited by 3,106
CategoryTheory.CartesianMonoidalCategorystatement and proof · cited by 947
CategoryTheory.SemiCartesianMonoidalCategory.sndproof · cited by 181
CategoryTheory.Limits.IsInitialstatement and proof · cited by 158
CategoryTheory.Limits.IsInitial.toproof · cited by 119
CategoryTheory.Closedstatement and proof · cited by 90

Cited by14

Results whose statement or proof uses this declaration.

CategoryTheory.mulZeroproof · cited by 4
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.isInitialIsTerminalIso_inv_leftstatement · cited by 1
CategoryTheory.zeroMul_homstatement and proof · cited by 1
CategoryTheory.strict_initialproof · cited by 1
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.rightUnitorproof · cited by 1
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.isInitialIso_hom_leftstatement · cited by 0
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.isInitialIso_inv_leftstatement · cited by 0
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.triangleproof · cited by 0
CategoryTheory.zeroMul_invstatement and proof · cited by 0
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.leftUnitor_naturalityproof · cited by 0
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.rightUnitor_naturalityproof · cited by 0
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.isInitialIsTerminalIso'_hom_leftstatement · cited by 0