Theorems · Theorem · category theory

CategoryTheory.Limits.IsColimit.eq_iff

∀ {J : Type u_1} {C : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} J]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} C] {FC : C → C → Type u_3} {CC : C → Type u_4}
  [inst_2 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_3 : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.PreservesColimitsOfShape J (CategoryTheory.forget C)] (F : CategoryTheory.Functor J C)
  [CategoryTheory.IsFilteredOrEmpty J] {t : CategoryTheory.Limits.Cocone F} (ht : CategoryTheory.Limits.IsColimit t)
  {i j : J} {xi : CategoryTheory.ToType (F.obj i)} {xj : CategoryTheory.ToType (F.obj j)},
  (CategoryTheory.ConcreteCategory.hom (t.ι.app i)) xi = (CategoryTheory.ConcreteCategory.hom (t.ι.app j)) xj ↔
    ∃ k f g, (CategoryTheory.ConcreteCategory.hom (F.map f)) xi = (CategoryTheory.ConcreteCategory.hom (F.map g)) xj

Defined in

Mathlib.CategoryTheory.Limits.ConcreteCategory.Filtered

Cited by

2 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.IsFilteredOrEmpty