Theorems · Theorem · category theory

CategoryTheory.IsVanKampenColimit.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] [CategoryTheory.Limits.HasColimitsOfShape J C]
  {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.IsVanKampenColimit 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]
      [∀ (X : C) (i : J) (f : X ⟶ c.pt),
          CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f (c.ι.app i)) Gl],
      CategoryTheory.IsVanKampenColimit (Gl.mapCocone c)

Defined in

Mathlib.CategoryTheory.Limits.VanKampen

Cited by

2 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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