Eckmann-Hilton argument #

The Eckmann-Hilton argument says that if a type carries two monoid structures that distribute over one another, then they are equal, and in addition commutative. The main application lies in proving that higher homotopy groups (πₙ for n ≥ 2) are commutative.

Main declarations #

eckmann_hilton.comm_monoid: If a type carries a unital magma structure that distributes over a unital binary operation, then the magma is a commutative monoid.
eckmann_hilton.comm_group: If a type carries a group structure that distributes over a unital binary operation, then the group is commutative.

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structure eckmann_hilton.is_unital {X : Type u} (m : X → X → X) (e : X) :

Prop

to_is_left_id : is_left_id X m e
to_is_right_id : is_right_id X m e

is_unital m e expresses that e : X is a left and right unit for the binary operation m : X → X → X.

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theorem eckmann_hilton.add_zero_class.is_unital {X : Type u} [G : add_zero_class X] :

eckmann_hilton.is_unital has_add.add 0

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theorem eckmann_hilton.mul_one_class.is_unital {X : Type u} [G : mul_one_class X] :

eckmann_hilton.is_unital has_mul.mul 1

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theorem eckmann_hilton.one {X : Type u} {m₁ m₂ : X → X → X} {e₁ e₂ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) (h₂ : eckmann_hilton.is_unital m₂ e₂) (distrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)) :

e₁ = e₂

If a type carries two unital binary operations that distribute over each other, then they have the same unit elements.

In fact, the two operations are the same, and give a commutative monoid structure, see eckmann_hilton.comm_monoid.

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theorem eckmann_hilton.mul {X : Type u} {m₁ m₂ : X → X → X} {e₁ e₂ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) (h₂ : eckmann_hilton.is_unital m₂ e₂) (distrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)) :

m₁ = m₂

If a type carries two unital binary operations that distribute over each other, then these operations are equal.

In fact, they give a commutative monoid structure, see eckmann_hilton.comm_monoid.

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theorem eckmann_hilton.mul_comm {X : Type u} {m₁ m₂ : X → X → X} {e₁ e₂ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) (h₂ : eckmann_hilton.is_unital m₂ e₂) (distrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)) :

is_commutative X m₂

If a type carries two unital binary operations that distribute over each other, then these operations are commutative.

In fact, they give a commutative monoid structure, see eckmann_hilton.comm_monoid.

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theorem eckmann_hilton.mul_assoc {X : Type u} {m₁ m₂ : X → X → X} {e₁ e₂ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) (h₂ : eckmann_hilton.is_unital m₂ e₂) (distrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)) :

is_associative X m₂

If a type carries two unital binary operations that distribute over each other, then these operations are associative.

In fact, they give a commutative monoid structure, see eckmann_hilton.comm_monoid.

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def eckmann_hilton.add_comm_monoid {X : Type u} {m₁ : X → X → X} {e₁ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) [h : add_zero_class X] (distrib : ∀ (a b c d : X), m₁ (a + b) (c + d) = m₁ a c + m₁ b d) :

add_comm_monoid X

If a type carries a unital additive magma structure that distributes over a unital binary operations, then the additive magma structure is a commutative additive monoid.

Equations

eckmann_hilton.add_comm_monoid h₁ distrib = {add := has_add.add (add_zero_class.to_has_add X), add_assoc := _, zero := 0, zero_add := _, add_zero := _, nsmul := add_monoid.nsmul._default 0 has_add.add add_zero_class.zero_add add_zero_class.add_zero, nsmul_zero' := _, nsmul_succ' := _, add_comm := _}

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def eckmann_hilton.comm_monoid {X : Type u} {m₁ : X → X → X} {e₁ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) [h : mul_one_class X] (distrib : ∀ (a b c d : X), m₁ (a * b) (c * d) = (m₁ a c) * m₁ b d) :

comm_monoid X

If a type carries a unital magma structure that distributes over a unital binary operations, then the magma structure is a commutative monoid.

Equations

eckmann_hilton.comm_monoid h₁ distrib = {mul := has_mul.mul (mul_one_class.to_has_mul X), mul_assoc := _, one := 1, one_mul := _, mul_one := _, npow := monoid.npow._default 1 has_mul.mul mul_one_class.one_mul mul_one_class.mul_one, npow_zero' := _, npow_succ' := _, mul_comm := _}

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def eckmann_hilton.comm_group {X : Type u} {m₁ : X → X → X} {e₁ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) [G : group X] (distrib : ∀ (a b c d : X), m₁ (a * b) (c * d) = (m₁ a c) * m₁ b d) :

comm_group X

If a type carries a group structure that distributes over a unital binary operation, then the group is commutative.

Equations

eckmann_hilton.comm_group h₁ distrib = {mul := comm_monoid.mul (eckmann_hilton.comm_monoid h₁ distrib), mul_assoc := _, one := comm_monoid.one (eckmann_hilton.comm_monoid h₁ distrib), one_mul := _, mul_one := _, npow := comm_monoid.npow (eckmann_hilton.comm_monoid h₁ distrib), npow_zero' := _, npow_succ' := _, inv := group.inv G, div := group.div G, div_eq_mul_inv := _, zpow := group.zpow G, zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, mul_left_inv := _, mul_comm := _}

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def eckmann_hilton.add_comm_group {X : Type u} {m₁ : X → X → X} {e₁ : X} (h₁ : eckmann_hilton.is_unital m₁ e₁) [G : add_group X] (distrib : ∀ (a b c d : X), m₁ (a + b) (c + d) = m₁ a c + m₁ b d) :

add_comm_group X

If a type carries an additive group structure that distributes over a unital binary operation, then the additive group is commutative.

Equations

eckmann_hilton.add_comm_group h₁ distrib = {add := add_comm_monoid.add (eckmann_hilton.add_comm_monoid h₁ distrib), add_assoc := _, zero := add_comm_monoid.zero (eckmann_hilton.add_comm_monoid h₁ distrib), zero_add := _, add_zero := _, nsmul := add_comm_monoid.nsmul (eckmann_hilton.add_comm_monoid h₁ distrib), nsmul_zero' := _, nsmul_succ' := _, neg := add_group.neg G, sub := add_group.sub G, sub_eq_add_neg := _, zsmul := add_group.zsmul G, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _}