Documentation

Mathlib.Algebra.Group.Action.End

Endomorphisms, homomorphisms and group actions #

This file defines Function.End as the monoid of endomorphisms on a type, and provides results relating group actions with these endomorphisms.

Notation #

Implementation details #

This file should avoid depending on other parts of GroupTheory, to avoid import cycles. More sophisticated lemmas belong in GroupTheory.GroupAction.

Tags #

group action

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@[reducible, inline]
abbrev Function.Surjective.addActionLeft {R : Type u_9} {S : Type u_10} {M : Type u_11} [AddMonoid R] [AddAction R M] [AddMonoid S] [VAdd S M] (f : R →+ S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c +ᵥ x = c +ᵥ x) :
AddAction S M

Push forward the action of R on M along a compatible surjective map f : R →+ S.

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@[reducible, inline]
abbrev Function.Surjective.mulActionLeft {R : Type u_9} {S : Type u_10} {M : Type u_11} [Monoid R] [MulAction R M] [Monoid S] [SMul S M] (f : R →* S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) :
MulAction S M

Push forward the action of R on M along a compatible surjective map f : R →* S.

See also Function.Surjective.distribMulActionLeft and Function.Surjective.moduleLeft.

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@[reducible, inline]
abbrev AddAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_5) [AddMonoid M] [AddAction M α] [AddMonoid N] (g : N →+ M) :
AddAction N α

An additive action of M on α and an additive monoid homomorphism N → M induce an additive action of N on α.

See note [reducible non-instances].

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@[reducible, inline]
abbrev MulAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_5) [Monoid M] [MulAction M α] [Monoid N] (g : N →* M) :
MulAction N α

A multiplicative action of M on α and a monoid homomorphism N → M induce a multiplicative action of N on α.

See note [reducible non-instances].

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theorem AddAction.compHom_vadd_def {E : Type u_9} {F : Type u_10} {G : Type u_11} [AddMonoid E] [AddMonoid F] [AddAction F G] (f : E →+ F) (a : E) (x : G) :
a +ᵥ x = f a +ᵥ x
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theorem MulAction.compHom_smul_def {E : Type u_9} {F : Type u_10} {G : Type u_11} [Monoid E] [Monoid F] [MulAction F G] (f : E →* F) (a : E) (x : G) :
a • x = f a • x
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def AddMonoidHom.vaddZeroHom {M : Type u_9} {N : Type u_10} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] :
M →+ N

If the additive action of M on N is compatible with addition on N, then fun x ↦ x +ᵥ 0 is an additive monoid homomorphism from M to N.

Equations
  • AddMonoidHom.vaddZeroHom = { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯, map_add' := ⋯ }
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@[simp]
theorem MonoidHom.smulOneHom_apply {M : Type u_9} {N : Type u_10} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] (x : M) :
MonoidHom.smulOneHom x = x • 1
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@[simp]
theorem AddMonoidHom.vaddZeroHom_apply {M : Type u_9} {N : Type u_10} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] (x : M) :
AddMonoidHom.vaddZeroHom x = x +ᵥ 0
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def MonoidHom.smulOneHom {M : Type u_9} {N : Type u_10} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] :
M →* N

If the multiplicative action of M on N is compatible with multiplication on N, then fun x ↦ x • 1 is a monoid homomorphism from M to N.

Equations
  • MonoidHom.smulOneHom = { toFun := fun (x : M) => x • 1, map_one' := ⋯, map_mul' := ⋯ }
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def addMonoidHomEquivAddActionIsScalarTower (M : Type u_9) (N : Type u_10) [AddMonoid M] [AddMonoid N] :
(M →+ N) ≃ { _inst : AddAction M N // VAddAssocClass M N N }

A monoid homomorphism between two additive monoids M and N can be equivalently specified by an additive action of M on N that is compatible with the addition on N.

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  • One or more equations did not get rendered due to their size.
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def monoidHomEquivMulActionIsScalarTower (M : Type u_9) (N : Type u_10) [Monoid M] [Monoid N] :
(M →* N) ≃ { _inst : MulAction M N // IsScalarTower M N N }

A monoid homomorphism between two monoids M and N can be equivalently specified by a multiplicative action of M on N that is compatible with the multiplication on N.

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  • One or more equations did not get rendered due to their size.
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def Function.End (α : Type u_5) :
Type u_5

The monoid of endomorphisms.

Note that this is generalized by CategoryTheory.End to categories other than Type u.

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instance instMonoidEnd (α : Type u_5) :
Monoid (Function.End α)
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instance instInhabitedEnd (α : Type u_5) :
Inhabited (Function.End α)
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instance Function.End.applyMulAction {α : Type u_5} :
MulAction (Function.End α) α

The tautological action by Function.End α on α.

This is generalized to bundled endomorphisms by:

  • Equiv.Perm.applyMulAction
  • AddMonoid.End.applyDistribMulAction
  • AddMonoid.End.applyModule
  • AddAut.applyDistribMulAction
  • MulAut.applyMulDistribMulAction
  • LinearEquiv.applyDistribMulAction
  • LinearMap.applyModule
  • RingHom.applyMulSemiringAction
  • RingAut.applyMulSemiringAction
  • AlgEquiv.applyMulSemiringAction
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@[simp]
theorem Function.End.smul_def {α : Type u_5} (f : Function.End α) (a : α) :
f • a = f a
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theorem Function.End.mul_def {α : Type u_5} (f : Function.End α) (g : Function.End α) :
f * g = f ∘ g
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theorem Function.End.one_def {α : Type u_5} :
1 = id
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def MulAction.toEndHom {M : Type u_1} {α : Type u_5} [Monoid M] [MulAction M α] :
M →* Function.End α

The monoid hom representing a monoid action.

When M is a group, see MulAction.toPermHom.

Equations
  • MulAction.toEndHom = { toFun := fun (x1 : M) (x2 : α) => x1 • x2, map_one' := ⋯, map_mul' := ⋯ }
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@[reducible, inline]
abbrev MulAction.ofEndHom {M : Type u_1} {α : Type u_5} [Monoid M] (f : M →* Function.End α) :
MulAction M α

The monoid action induced by a monoid hom to Function.End α

See note [reducible non-instances].

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