Theorems · Theorem · category theory

CategoryTheory.MorphismProperty.isLocal_single_iff_bijective

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) (Z : C),
  (CategoryTheory.MorphismProperty.single f).isLocal Z ↔
    Function.Bijective fun g => CategoryTheory.CategoryStruct.comp f g

Defined in: Mathlib.CategoryTheory.ObjectProperty.Local
Cited by: 1 results in Mathlib
Foundations: Depth 7 from the axioms · uses no axioms
Assumes: CategoryTheory.Category

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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
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
Function.Bijectivestatement and proof · cited by 863
CategoryTheory.MorphismProperty.ofHoms.casesOnproof · cited by 26
CategoryTheory.MorphismProperty.isLocalstatement and proof · cited by 22
CategoryTheory.MorphismProperty.singlestatement and proof · cited by 7

Cited by1

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CategoryTheory.Presheaf.nonempty_isLimit_mapCone_iffproof · cited by 1