Theorems · Theorem · category theory

CategoryTheory.projective_of_preservesFiniteColimits_preadditiveCoyonedaObj

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (P : C)
  [hP : CategoryTheory.Limits.PreservesFiniteColimits (CategoryTheory.preadditiveCoyonedaObj P)],
  CategoryTheory.Projective P

An object is projective if its preadditive co-Yoneda functor preserves finite colimits.

Defined in: Mathlib.CategoryTheory.Abelian.Projective.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Abelian CategoryTheory.Limits.PreservesFiniteColimits

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CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Abelianstatement and proof · cited by 1,753
ModuleCatstatement · cited by 1,429
MulOppositestatement · cited by 1,135
CategoryTheory.Endstatement · cited by 169
CategoryTheory.Limits.PreservesFiniteColimitsstatement and proof · cited by 102
CategoryTheory.Projectivestatement · cited by 78
CategoryTheory.Functor.PreservesHomologyproof · cited by 42
CategoryTheory.preadditiveCoyonedaObjstatement and proof · cited by 11
CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyonedaObjproof · cited by 1

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