Theorems · Definition · category theory

CategoryTheory.Limits.evaluationJointlyReflectsLimits

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : Type u₁} →
      [inst_1 : CategoryTheory.Category.{v₁, u₁} J] →
        {K : Type u₂} →
          [inst_2 : CategoryTheory.Category.{v₂, u₂} K] →
            {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} →
              (c : CategoryTheory.Limits.Cone F) →
                ((k : K) → CategoryTheory.Limits.IsLimit (((CategoryTheory.evaluation K C).obj k).mapCone c)) →
                  CategoryTheory.Limits.IsLimit c

The evaluation functors jointly reflect limits: that is, to show a cone is a limit of F it suffices to show that each evaluation cone is a limit. In other words, to prove a cone is limiting you can show it's pointwise limiting.

Defined in

Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Cited by

7 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category