Theorems · Theorem · category theory

CategoryTheory.IsUniversalColimit.map_reflective

∀ {J : Type v'} [inst : CategoryTheory.Category.{u', v'} J] {C : Type u} [inst_1 : CategoryTheory.Category.{v, u} C]
  {D : Type u_2} [inst_2 : CategoryTheory.Category.{v_2, u_2} D] {Gl : CategoryTheory.Functor C D}
  {Gr : CategoryTheory.Functor D C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful] {F : CategoryTheory.Functor J D}
  {c : CategoryTheory.Limits.Cocone (F.comp Gr)},
  CategoryTheory.IsUniversalColimit c →
    ∀ [∀ (X : D) (f : X ⟶ Gl.obj c.pt), CategoryTheory.Limits.HasPullback (Gr.map f) (adj.unit.app c.pt)]
      [∀ (X : D) (f : X ⟶ Gl.obj c.pt),
          CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan (Gr.map f) (adj.unit.app c.pt)) Gl],
      CategoryTheory.IsUniversalColimit (Gl.mapCocone c)

Defined in

Mathlib.CategoryTheory.Limits.VanKampen

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.Limits.HasPullbackCategoryTheory.Limits.PreservesLimit