Theorems · Theorem · category theory

CategoryTheory.Limits.Concrete.Pi.map_ext

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} (f : J → C)
  [inst_1 : CategoryTheory.Limits.HasProduct f] {D : Type t} [inst_2 : CategoryTheory.Category.{r, t} D]
  {FD : D → D → Type u_1} {DD : D → Type (max w r)} [inst_3 : (X Y : D) → FunLike (FD X Y) (DD X) (DD Y)]
  [inst_4 : CategoryTheory.ConcreteCategory D FD] (F : CategoryTheory.Functor C D)
  [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Discrete.functor f) F]
  [CategoryTheory.Limits.HasProduct fun j => F.obj (f j)]
  [CategoryTheory.Limits.PreservesLimitsOfShape CategoryTheory.Limits.WalkingCospan (CategoryTheory.forget D)]
  [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Discrete.functor fun b => F.obj (f b))
      (CategoryTheory.forget D)]
  (x y : CategoryTheory.ToType (F.obj (∏ᶜ f))),
  (∀ (i : J),
      (CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.Limits.Pi.π f i))) x =
        (CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.Limits.Pi.π f i))) y) →
    x = y

Defined in

Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Cited by

0 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasProductCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesLimitCategoryTheory.Limits.HasProductCategoryTheory.Limits.PreservesLimitsOfShapeCategoryTheory.Limits.PreservesLimit