mathlib documentation

category_theory.endomorphism

Endomorphisms #

Definition and basic properties of endomorphisms and automorphisms of an object in a category.

For each X : C, we provide End X := X ⟶ X with a monoid structure, and Aut X := X ≅ X with a group structure.

@[protected, instance]

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

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Assist the typechecker by expressing a morphism X ⟶ X as a term of End X.

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Assist the typechecker by expressing an endomorphism f : End X as a term of X ⟶ X.

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

Endomorphisms of an object form a monoid

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@[protected, instance]
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theorem category_theory.End.smul_right {C : Type u} [category_theory.category C] {X Y : C} {r : category_theory.End Y} {f : X Y} :
r f = f r
def category_theory.Aut {C : Type u} [category_theory.category C] (X : C) :

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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Instances for category_theory.Aut

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

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Isomorphisms induce isomorphisms of the automorphism group

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f.map as a monoid hom between endomorphism monoids.

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@[simp]
theorem category_theory.functor.map_End_apply {C : Type u} [category_theory.category C] (X : C) {D : Type u'} [category_theory.category D] (f : C D) (ᾰ : X X) :

f.map_iso as a group hom between automorphism groups.

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