Theorems · Theorem · category theory

CategoryTheory.strict_initial

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C] {A : C}
  [CategoryTheory.Closed A] {I : C} (t : CategoryTheory.Limits.IsInitial I) (f : A ⟶ I), CategoryTheory.IsIso f

If an initial object I exists in a CCC then it is a strict initial object, i.e. any morphism to I is an iso. This actually shows a slightly stronger version: any morphism to an initial object from an exponentiable object is an isomorphism.

Defined in

Mathlib.CategoryTheory.Monoidal.Closed.Cartesian

Cited by

1 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.Closed