mathlib docs: category_theory.endomorphism

def category_theory.End {C : Type u} [category_theory.category_struct C] (X : C) :

Type v

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

Equations

category_theory.End X = (X ⟶ X)

source

@[instance]

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

has_one (category_theory.End X)

Equations

category_theory.End.has_one X = {one := 𝟙 X}

source

@[instance]

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

has_mul (category_theory.End X)

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

Equations

category_theory.End.has_mul X = {mul := λ (x y : category_theory.End X), y ≫ x}

source

@[simp]

theorem category_theory.End.one_def {C : Type u} [category_theory.category_struct C] {X : C} :

1 = 𝟙 X

source

@[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

source

@[instance]

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

monoid (category_theory.End X)

Endomorphisms of an object form a monoid

Equations

category_theory.End.monoid = {mul := has_mul.mul (category_theory.End.has_mul X), mul_assoc := _, one := 1, one_mul := _, mul_one := _}

source

@[instance]

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

group (category_theory.End X)

In a groupoid, endomorphisms form a group

Equations

category_theory.End.group X = {mul := monoid.mul category_theory.End.monoid, mul_assoc := _, one := monoid.one category_theory.End.monoid, one_mul := _, mul_one := _, inv := category_theory.groupoid.inv X, mul_left_inv := _}

source

def category_theory.Aut {C : Type u} [category_theory.category C] (X : C) :

Type v

Equations

category_theory.Aut X = (X ≅ X)

source

@[instance]

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

group (category_theory.Aut X)

Equations

category_theory.Aut.group X = {mul := flip category_theory.iso.trans, mul_assoc := _, one := category_theory.iso.refl X, one_mul := _, mul_one := _, inv := category_theory.iso.symm X, mul_left_inv := _}

source

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

units (category_theory.End X) ≃* category_theory.Aut X

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

Equations

category_theory.Aut.units_End_equiv_Aut X = {to_fun := λ (f : units (category_theory.End X)), {hom := f.val, inv := f.inv, hom_inv_id' := _, inv_hom_id' := _}, inv_fun := λ (f : category_theory.Aut X), {val := f.hom, inv := f.inv, val_inv := _, inv_val := _}, left_inv := _, right_inv := _, map_mul' := _}

source

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

category_theory.End X →* category_theory.End (f.obj X)

f.map as a monoid hom between endomorphism monoids.

Equations

category_theory.functor.map_End X f = {to_fun := f.map X, map_one' := _, map_mul' := _}

source

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

category_theory.Aut X →* category_theory.Aut (f.obj X)

f.map_iso as a group hom between automorphism groups.

Equations

category_theory.functor.map_Aut X f = {to_fun := f.map_iso X, map_one' := _, map_mul' := _}