Theorems · Theorem · category theory

CategoryTheory.Functor.projective_obj_of_projective

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  (F : CategoryTheory.Functor C D) [F.PreservesProjectiveObjects] {X : C},
  CategoryTheory.Projective X → CategoryTheory.Projective (F.obj X)

See Functor.projective_obj for a variant taking Projective X as a typeclass argument.

Defined in: Mathlib.CategoryTheory.Preadditive.Projective.Preserves
Cited by: 1 results in Mathlib
Foundations: Depth 4 from the axioms · uses no axioms
Assumes: CategoryTheory.Category CategoryTheory.Category CategoryTheory.Functor.PreservesProjectiveObjects

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Functor.objstatement · cited by 19,642
CategoryTheory.Functorstatement and proof · cited by 16,252
CategoryTheory.Projectivestatement and proof · cited by 78
CategoryTheory.Functor.PreservesProjectiveObjectsstatement and proof · cited by 7
CategoryTheory.Functor.PreservesProjectiveObjects.projective_objproof · cited by 2

Cited by1

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ModuleCat.hasProjectiveDimensionLE_iff_forall_maximalSpectrumproof · cited by 2