Theorems · Theorem · category theory

CategoryTheory.Functor.preservesFiniteLimits_of_preservesKernels

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Preadditive C] {D : Type u₂}
  [inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D)
  [F.PreservesZeroMorphisms] [CategoryTheory.Limits.HasBinaryBiproducts C] [CategoryTheory.Limits.HasFiniteProducts C]
  [CategoryTheory.Limits.HasEqualizers C] [CategoryTheory.Limits.HasZeroObject C]
  [CategoryTheory.Limits.HasZeroObject D]
  [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) F],
  CategoryTheory.Limits.PreservesFiniteLimits F

A functor between preadditive categories which preserves kernels preserves all finite limits.

Defined in

Mathlib.CategoryTheory.Preadditive.LeftExact

Cited by

2 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasBinaryBiproductsCategoryTheory.Limits.HasFiniteProductsCategoryTheory.Limits.HasEqualizersCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.PreservesLimit