Theorems · Definition · category theory

CategoryTheory.unitCompPartialBijective

{C : Type u₁} →
  {D : Type u₂} →
    [inst : CategoryTheory.Category.{v₁, u₁} C] →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {i : CategoryTheory.Functor D C} →
          [inst_2 : CategoryTheory.Reflective i] →
            (A : C) → {B : C} → i.essImage B → (A ⟶ B) ≃ (i.obj ((CategoryTheory.reflector i).obj A) ⟶ B)

If i has a reflector L, then the function (i.obj (L.obj A) ⟶ B) → (A ⟶ B) given by precomposing with η.app A is a bijection provided B is in the essential image of i. That is, the function fun (f : i.obj (L.obj A) ⟶ B) ↦ η.app A ≫ f is bijective, as long as B is in the essential image of i. This definition gives an equivalence: the key property that the inverse can be described nicely is shown in unitCompPartialBijective_symm_apply. This establishes there is a natural bijection (A ⟶ B) ≃ (i.obj (L.obj A) ⟶ B). In other words, from the point of view of objects in D, A and i.obj (L.obj A) look the same: specifically that η.app A is an isomorphism.

Defined in

Mathlib.CategoryTheory.Adjunction.Reflective

Cited by

6 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Reflective