# mathlibdocumentation

category_theory.endomorphism

def category_theory.End {C : Type u} (X : C) :
Type v

Endomorphisms of an object in a category. Arguments order in multiplication agrees with function.comp, not with category.comp.

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@[instance]
def category_theory.End.has_one {C : Type u} (X : C) :

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@[instance]
def category_theory.End.inhabited {C : Type u} (X : C) :

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@[instance]
def category_theory.End.has_mul {C : Type u} (X : C) :

Multiplication of endomorphisms agrees with function.comp, not category_struct.comp.

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@[simp]
theorem category_theory.End.one_def {C : Type u} {X : C} :
1 = 𝟙 X

@[simp]
theorem category_theory.End.mul_def {C : Type u} {X : C} (xs ys : category_theory.End X) :
xs * ys = ys xs

@[instance]
def category_theory.End.monoid {C : Type u} {X : C} :

Endomorphisms of an object form a monoid

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@[instance]
def category_theory.End.group {C : Type u} (X : C) :

In a groupoid, endomorphisms form a group

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def category_theory.Aut {C : Type u} (X : C) :
Type v

Automorphisms of an object in a category.

The order of arguments in multiplication agrees with function.comp, not with category.comp.

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@[instance]
def category_theory.Aut.inhabited {C : Type u} (X : C) :

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@[instance]
def category_theory.Aut.group {C : Type u} (X : C) :

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def category_theory.Aut.units_End_equiv_Aut {C : Type u} (X : C) :

Units in the monoid of endomorphisms of an object are (multiplicatively) equivalent to automorphisms of that object.

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def category_theory.functor.map_End {C : Type u} (X : C) {D : Type u'} (f : C D) :

f.map as a monoid hom between endomorphism monoids.

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def category_theory.functor.map_Aut {C : Type u} (X : C) {D : Type u'} (f : C D) :

f.map_iso as a group hom between automorphism groups.

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