File: Mathlib/CategoryTheory/MorphismProperty/Concrete.lean

full_name: CategoryTheory.MorphismProperty.injective
code: /-- Injectiveness (in a concrete category) as a `MorphismProperty` -/
protected def injective : MorphismProperty C 
goal: CategoryTheory.MorphismProperty C

full_name: CategoryTheory.MorphismProperty.surjective
code: /-- Surjectiveness (in a concrete category) as a `MorphismProperty` -/
protected def surjective : MorphismProperty C 
goal: CategoryTheory.MorphismProperty C

full_name: CategoryTheory.MorphismProperty.bijective
code: /-- Bijectiveness (in a concrete category) as a `MorphismProperty` -/
protected def bijective : MorphismProperty C 
goal: CategoryTheory.MorphismProperty C

full_name: CategoryTheory.MorphismProperty.bijective_eq_sup
code: theorem bijective_eq_sup :
    MorphismProperty.bijective C = MorphismProperty.injective C ⊓ MorphismProperty.surjective C 
goal: CategoryTheory.MorphismProperty.bijective C =
  CategoryTheory.MorphismProperty.injective C ⊓ CategoryTheory.MorphismProperty.surjective C

full_name: CategoryTheory.MorphismProperty.instIsMultiplicativeInjective
code: instance : (MorphismProperty.injective C).IsMultiplicative 
goal: (CategoryTheory.MorphismProperty.injective C).IsMultiplicative

full_name: CategoryTheory.MorphismProperty.instIsMultiplicativeSurjective
code: instance : (MorphismProperty.surjective C).IsMultiplicative 
goal: (CategoryTheory.MorphismProperty.surjective C).IsMultiplicative

full_name: CategoryTheory.MorphismProperty.instIsMultiplicativeBijective
code: instance : (MorphismProperty.bijective C).IsMultiplicative 
goal: (CategoryTheory.MorphismProperty.bijective C).IsMultiplicative

full_name: CategoryTheory.MorphismProperty.injective_respectsIso
code: instance injective_respectsIso : (MorphismProperty.injective C).RespectsIso 
goal: (CategoryTheory.MorphismProperty.injective C).RespectsIso

full_name: CategoryTheory.MorphismProperty.surjective_respectsIso
code: instance surjective_respectsIso : (MorphismProperty.surjective C).RespectsIso 
goal: (CategoryTheory.MorphismProperty.surjective C).RespectsIso

full_name: CategoryTheory.MorphismProperty.bijective_respectsIso
code: instance bijective_respectsIso : (MorphismProperty.bijective C).RespectsIso 
goal: (CategoryTheory.MorphismProperty.bijective C).RespectsIso

full_name: CategoryTheory.ConcreteCategory.HasSurjectiveInjectiveFactorization
code: /-- The property that any morphism in a concrete category can be factored as a surjective
map followed by an injective map. -/
abbrev HasSurjectiveInjectiveFactorization 
goal: Prop

full_name: CategoryTheory.ConcreteCategory.HasFunctorialSurjectiveInjectiveFactorization
code: /-- The property that any morphism in a concrete category can be functorially
factored as a surjective map followed by an injective map. -/
abbrev HasFunctorialSurjectiveInjectiveFactorization 
goal: Prop

full_name: CategoryTheory.ConcreteCategory.FunctorialSurjectiveInjectiveFactorizationData
code: /-- The structure containing the data of a functorial factorization of morphisms as
a surjective map followed by an injective map in a concrete category. -/
abbrev FunctorialSurjectiveInjectiveFactorizationData 
goal: Type (max u v)

full_name: CategoryTheory.functorialSurjectiveInjectiveFactorizationData
code: /-- In the category of types, any map can be functorially factored as a surjective
map followed by an injective map. -/
def functorialSurjectiveInjectiveFactorizationData :
    FunctorialSurjectiveInjectiveFactorizationData (Type u) 
goal: CategoryTheory.ConcreteCategory.FunctorialSurjectiveInjectiveFactorizationData (Type u)

full_name: CategoryTheory.instHasFunctorialSurjectiveInjectiveFactorizationType
code: instance : HasFunctorialSurjectiveInjectiveFactorization (Type u) 
goal: CategoryTheory.ConcreteCategory.HasFunctorialSurjectiveInjectiveFactorization (Type u)