Theorems · Definition · category theory

CategoryTheory.Limits.Concrete.prodEquiv

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {FC : C → C → Type u_1} →
      {CC : C → Type w} →
        [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] →
          [inst_2 : CategoryTheory.ConcreteCategory C FC] →
            (X₁ X₂ : C) →
              [inst_3 : CategoryTheory.Limits.HasBinaryProduct X₁ X₂] →
                [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.pair X₁ X₂) (CategoryTheory.forget C)] →
                  CategoryTheory.ToType (X₁ ⨯ X₂) ≃ CategoryTheory.ToType X₁ × CategoryTheory.ToType X₂

The equivalence ToType (X₁ ⨯ X₂) ≃ (ToType X₁) × (ToType X₂) if X₁ and X₂ are objects in a concrete category C.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Cited by

5 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasBinaryProductCategoryTheory.Limits.PreservesLimit