Theorems · Theorem · category theory

CategoryTheory.Adjunction.hasExactLimitsOfShape

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u''} [inst_1 : CategoryTheory.Category.{v'', u''} D]
  {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.Full] [F.Faithful] (J : Type u')
  [inst_4 : CategoryTheory.Category.{v', u'} J] [inst_5 : CategoryTheory.Limits.HasLimitsOfShape J C]
  [inst_6 : CategoryTheory.Limits.HasLimitsOfShape J D] [CategoryTheory.HasExactLimitsOfShape J D]
  [CategoryTheory.Limits.HasFiniteColimits C] [CategoryTheory.Limits.PreservesFiniteColimits G],
  CategoryTheory.HasExactLimitsOfShape J C

Let adj : F ⊣ G be an adjunction, with F : C ⥤ D coreflective. Assume that C has finite colimits and G commutes with them. If D has exact limits of shape J, then C also has exact limits of shape J.

Defined in

Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.CategoryCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.HasExactLimitsOfShapeCategoryTheory.Limits.HasFiniteColimitsCategoryTheory.Limits.PreservesFiniteColimits