Theorems · Definition · category theory

CategoryTheory.cartesianClosedOfReflective

{C : Type u₁} →
  {D : Type u₂} →
    [inst : CategoryTheory.Category.{v₁, u₁} C] →
      [inst_1 : CategoryTheory.Category.{v₁, u₂} D] →
        (i : CategoryTheory.Functor D C) →
          [inst_2 : CategoryTheory.CartesianMonoidalCategory C] →
            [CategoryTheory.Reflective i] →
              [inst_4 : CategoryTheory.MonoidalClosed C] →
                [inst_5 : CategoryTheory.CartesianMonoidalCategory D] →
                  [CategoryTheory.ExponentialIdeal i] → CategoryTheory.MonoidalClosed D

If i witnesses that D is a reflective subcategory and an exponential ideal, then D is itself Cartesian closed. Unlike cartesianClosedOfReflective' this construction lifts exponential objects in C to exponential objects in D by applying the reflector to them, even though they already lie in the essential image of i; if you need better control over definitional equality, use cartesianClosedOfReflective' instead.

Defined in

Mathlib.CategoryTheory.Monoidal.Closed.Ideal

Cited by

0 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.ReflectiveCategoryTheory.MonoidalClosedCategoryTheory.CartesianMonoidalCategoryCategoryTheory.ExponentialIdeal