Theorems · Definition · category theory

CategoryTheory.Limits.initialMul

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [CategoryTheory.Limits.HasStrictInitialObjects C] →
      [inst_2 : CategoryTheory.Limits.HasInitial C] →
        (X : C) → [inst_3 : CategoryTheory.Limits.HasBinaryProduct (⊥_ C) X] → (⊥_ C) ⨯ X ≅ ⊥_ C

The product of X with an initial object in a category with strict initial objects is itself initial. This is the generalisation of the fact that Empty × X ≃ Empty for types (or 0 * n = 0).

Defined in

Mathlib.CategoryTheory.Limits.Shapes.StrictInitial

Cited by

2 results in Mathlib

Foundations

Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasStrictInitialObjectsCategoryTheory.Limits.HasInitialCategoryTheory.Limits.HasBinaryProduct