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Theorems · Theorem · category theory

CategoryTheory.Functor.preservesFiniteColimits_of_preservesHomology

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Preadditive C]
  [inst_3 : CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [inst_4 : F.Additive] [F.PreservesHomology]
  [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasFiniteCoproducts C]
  [CategoryTheory.Limits.HasCokernels C], CategoryTheory.Limits.PreservesFiniteColimits F

An additive which preserves homology preserves finite colimits.

Defined in
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
Cited by
2 results in Mathlib
Foundations
Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Functor.PreservesHomologyCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.HasFiniteCoproductsCategoryTheory.Limits.HasCokernels
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

CategoryTheory.Functor.exact_tfae · cited by 3Functor.exact_tfaeCategoryTheory.hasExactLimitsOfShape_of_preservesEpi · cited by 0CategoryTheory.hasExactLi…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.Functor.obj · cited by 19642Functor.objCategoryTheory.Functor · cited by 16252CategoryTheory.FunctorCategoryTheory.Functor.map · cited by 8698Functor.mapCategoryTheory.Preadditive · cited by 3309CategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObject · cited by 1298Limits.HasZeroObjectCategoryTheory.Functor.Additive · cited by 1179Functor.AdditiveCategoryTheory.Limits.parallelPair · cited by 766Limits.parallelPairCategoryTheory.Functor.map_id · cited by 616Functor.map_idCategoryTheory.Limits.PreservesColimit · cited by 278Limits.PreservesColimitCategoryTheory.Limits.HasBinaryBiproducts · cited by 165Limits.HasBinaryBiproductsCategoryTheory.Limits.HasFiniteCoproducts · cited by 110Limits.HasFiniteCoproductsCategoryTheory.Limits.PreservesFiniteColimits · cited by 102Limits.PreservesFiniteCol…CategoryTheory.Functor.map_zero · cited by 85Functor.map_zeroFunctor.preservesFiniteColimi…CITED BYCITES

Cites24

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Cited by2

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