The Eckmann-Hilton argument

Content created by Egbert Rijke, Fredrik Bakke and Raymond Baker.

Created on 2023-09-24.
Last modified on 2024-03-13.

module synthetic-homotopy-theory.eckmann-hilton-argument where
Imports 
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.interchange-law
open import foundation.path-algebra
open import foundation.universe-levels
open import foundation.whiskering-identifications-concatenation

open import structured-types.pointed-equivalences
open import structured-types.pointed-types

open import synthetic-homotopy-theory.double-loop-spaces
open import synthetic-homotopy-theory.functoriality-loop-spaces
open import synthetic-homotopy-theory.loop-spaces
open import synthetic-homotopy-theory.triple-loop-spaces

Idea

There are two classical statements of the Eckmann-Hilton argument. The first states that a group object in the category of groups is abelian. The second states that π₂(X) is abelian, for any space X. The former is an algebraic statement, while the latter is a homotopy theoretic statment. As it turns out, the two are equivalent. See the following Wikipedia article.

Both of these phrasings, however, are about set level structures. Since we have access to untruncated types, it is more natural to consider untruncated analogs of the above two statements. Thus, we will work with the following statement of the Eckmann-Hilton argument:

  (α β : Ω² X) → α ∙ β = β ∙ α

For fixed 2-loops, we will call the resulting identification "the Eckmann-Hilton identification". In this file we will give two different constructions of this identification, one that corresponds to the more algebraic statement and one that corresponds to the more homotopy theoretic statement. We will call the constructions themselves "the Eckmann-Hilton argument".

Definitions

Constructing the Eckmann-Hilton identification from the interchange law

The more algebraic argument uses the interchange law interchange-Ω². The interchange law essentially expresses that horizontal-concat-Ω² is a group homomorphism of vertical-concat-Ω² in each variable.

module _
  {l : Level} {A : UU l} {a : A}
  where

  outer-eckmann-hilton-interchange-connection-Ω² :
    (α δ : type-Ω² a) →
    horizontal-concat-Ω² α δ ＝ vertical-concat-Ω² α δ
  outer-eckmann-hilton-interchange-connection-Ω² α δ =
    ( z-concat-Id³ {α = α} {γ = δ} (inv right-unit) (inv left-unit)) ∙
    ( ( interchange-Ω² α refl refl δ) ∙
      ( y-concat-Id³ {β = α} {δ = δ}
        ( right-unit-law-horizontal-concat-Ω² {α = α})
        ( left-unit-law-horizontal-concat-Ω² {α = δ})))

  inner-eckmann-hilton-interchange-connection-Ω² :
    (β γ : type-Ω² a) → horizontal-concat-Ω² β γ ＝ vertical-concat-Ω² γ β
  inner-eckmann-hilton-interchange-connection-Ω² β γ =
    ( z-concat-Id³ {α = β} {β} {γ} (inv left-unit) (inv right-unit)) ∙
    ( ( interchange-Ω² refl β γ refl) ∙
      ( y-concat-Id³ {β = γ} {δ = β}
        ( left-unit-law-horizontal-concat-Ω² {α = γ})
        ( right-unit-law-horizontal-concat-Ω² {α = β})))

  eckmann-hilton-interchange-Ω² : (α β : type-Ω² a) → α ∙ β ＝ β ∙ α
  eckmann-hilton-interchange-Ω² α β =
    ( inv (outer-eckmann-hilton-interchange-connection-Ω² α β)) ∙
    ( inner-eckmann-hilton-interchange-connection-Ω² α β)

  interchange-concat-Ω² :
    (α β γ δ : type-Ω² a) → (α ∙ β) ∙ (γ ∙ δ) ＝ (α ∙ γ) ∙ (β ∙ δ)
  interchange-concat-Ω² =
    interchange-law-commutative-and-associative
      ( _∙_)
      ( eckmann-hilton-interchange-Ω²)
      ( assoc)

Constructing the Eckmann-Hilton identification using the naturality condition of the operation of whiskering a fixed 2-path by a 1-path

The motivation

Now we give the more homotopy theoretic version of the Eckmann-Hilton argument. Consider 2-loops α β : Ω² (X , base). The more homotopy theoretic Eckmann-Hilton argument is often depicted as follows:

| α |      | refl-Ω² | α |      | β | refl-Ω² |       | β |
-----  ＝  ----------------  ＝  ----------------  ＝  ----
| β |      | β | refl-Ω² |      | refl-Ω² | α |       | α |

The first picture represents the vertical concatenation of α and β. The notation | γ | δ | represents the horizontal concatenation of 2-dimensional identifications γ and δ. Then | refl | α | is just left-whisker-concat refl-Ω² α. The first and last equality come from the unit laws of whiskering. And the middle equality can be recognized as commutative-left-whisker-right-whisker-concat, which is the naturality condition of left-whisker-concat - α applied to β.

Since this version of the Eckmann-Hilton argument may seem more complicated than the algbraic version, the reader is entitled to wonder why we bother giving this second version.

This version of the Eckmann-Hilton argument makes more salient the connection between the Eckmann-Hilton argument and the 2-D descent data of a fibration. For now, consider the family of based path spaces Id base : X → UU. A 1-loop l induces an autoequivalence Ω X ≃ Ω X given by concatinating on the right by l. This is shown in tr-Id-right. A 2-loop s induces a homotpy id {A = Ω X} ~ id given by left-whisker-concat. This claim is shown in TODO (provide link). Thus, the 2-D descent data of Id base is (up to equivalence) exactly the homotopy at the heart of this version of the Eckmann-Hilton argument.

Recall that homotpies of type id ~ id automatically commute with each other via eckmann-hilton-htpy. This identification is constructed using the naturality condition of the two homotopies involved. Thus, in the case of Id base, we can see a very close correspondence between the Eckmann-Hilton identification of 2-loops in the base type X and the Eckmann-Hilton identification of the homotopies induced by said 2-loops.

Of course Id base is a special type family. But this idea generalizes nonetheless. Given a type family B : X → UU, any 2-loops α β : Ω X induce homotopies tr² B α and tr² B β of type id {A = B base} ~ id. Again, these homotpies automatically commute with each other via their naturality conditions. Then, the naturality condition that makes α and β commute in Ω² X is sent by tr³ B to the naturality condition that makes the induced homotopies commute. This is recorded in tr³-htpy-swap-path-swap. From this, it is easy to show that "transport preserves the Eckmann-Hilton identification" by proving that the additional coherence paths in the definition of eckmann-hilton and eckmann-hilton-htpy are compatible.

This connection has important consequences, one of which being the connection between the Eckmann-Hilton argument and the Hopf fibration.

The construction, using left whiskering

module _
  {l : Level} {A : Pointed-Type l}
  where

  eckmann-hilton-Ω² :
    (α β : type-Ω² (point-Pointed-Type A)) → α ∙ β ＝ β ∙ α
  eckmann-hilton-Ω² α β =
    ( inv
      ( horizontal-concat-Id²
        ( left-unit-law-left-whisker-Ω² α)
        ( right-unit-law-right-whisker-Ω² β))) ∙
    ( commutative-left-whisker-right-whisker-concat α β) ∙
    ( horizontal-concat-Id²
      ( right-unit-law-right-whisker-Ω² β)
      ( left-unit-law-left-whisker-Ω² α))

Using right whiskering

There is another natural construction of an Eckmann-Hilton identification along these lines. If we think of the first construction as "rotating clockwise", this alternate version "rotates counter-clockwise". In terms of braids, the previous construction of Eckmann-Hilton braids α over β, while this new construction braids α under β. This difference shows up nicely in the type theory. The first version uses the naturality of the operation of whiskering on the left, while the second version uses the naturality of the operation of whiskering on the right. Based on the intution of braiding, we should expect these two version of the Eckmann-Hilton identification to naturally "undo" each other, which they do. Thus, we will refer to this alternate construction of Eckmann-Hilton as "the inverse Eckmann-Hilton argument", and the corresponding identification "the inverse Eckmann-Hilton identification".

module _
  {l : Level} {A : Pointed-Type l}
  where

  inv-eckmann-hilton-Ω² :
    (α β : type-Ω² (point-Pointed-Type A)) → α ∙ β ＝ β ∙ α
  inv-eckmann-hilton-Ω² α β =
    ( inv
      ( horizontal-concat-Id²
        ( right-unit-law-right-whisker-Ω² α)
        ( left-unit-law-left-whisker-Ω² β))) ∙
    ( commutative-right-whisker-left-whisker-concat α β) ∙
    ( horizontal-concat-Id²
      ( left-unit-law-left-whisker-Ω² β)
      ( right-unit-law-right-whisker-Ω² α))

We now prove that this Eckmann-Hilton identification "undoes" the previously constructed Eckmann-Hilton identification. If we think of braiding α over β, then braiding β under α, we should end up with the trivial braid. Thus, we should have

eckmann-hilton-Ω² α β ∙ inv-eckmann-hilton-Ω² β α ＝ refl

This is equivalent to,

inv inv-eckmann-hilton-Ω² β α ＝ eckmann-hilton-Ω² α β

which is what we prove.

Note. that the above property is distinct from syllepsis, since it concerns two different construction of the Eckmann-Hilton identification. Further, it works for all 2-loops, not just 3-loops.

module _
  {l : Level} {A : Pointed-Type l}
  where

  compute-inv-inv-eckmann-hilton-Ω² :
    (α β : type-Ω² (point-Pointed-Type A)) →
    inv (inv-eckmann-hilton-Ω² β α) ＝ eckmann-hilton-Ω² α β
  compute-inv-inv-eckmann-hilton-Ω² α β =
    ( distributive-inv-concat
      ( ( inv
          ( horizontal-concat-Id²
            ( right-unit-law-right-whisker-Ω² β)
            ( left-unit-law-left-whisker-Ω² α))) ∙
        ( commutative-right-whisker-left-whisker-concat β α))
      ( horizontal-concat-Id²
        ( left-unit-law-left-whisker-Ω² α)
        ( right-unit-law-right-whisker-Ω² β))) ∙
    ( left-whisker-concat
      ( inv
        ( horizontal-concat-Id²
          ( left-unit-law-left-whisker-Ω² α)
            ( right-unit-law-right-whisker-Ω² β)))
      ( distributive-inv-concat
        ( inv
          ( horizontal-concat-Id²
            ( right-unit-law-right-whisker-Ω² β)
            ( left-unit-law-left-whisker-Ω² α)))
        ( commutative-right-whisker-left-whisker-concat β α))) ∙
    ( left-whisker-concat
      ( inv
        ( horizontal-concat-Id²
          ( left-unit-law-left-whisker-Ω² α)
            ( right-unit-law-right-whisker-Ω² β)))
      ( horizontal-concat-Id²
        ( compute-inv-commutative-left-whisker-right-whisker-concat α β)
        ( inv-inv
          ( horizontal-concat-Id²
            ( right-unit-law-right-whisker-Ω² β)
            ( left-unit-law-left-whisker-Ω² α))))) ∙
    ( inv
      ( assoc
        ( inv
          ( horizontal-concat-Id²
            ( left-unit-law-left-whisker-Ω² α)
            ( right-unit-law-right-whisker-Ω² β)))
        ( commutative-left-whisker-right-whisker-concat α β)
        ( horizontal-concat-Id²
          ( right-unit-law-right-whisker-Ω² β)
          ( left-unit-law-left-whisker-Ω² α))))

Properties

We can apply each eckmann-hilton-Ω² and inv-eckmann-hilton-Ω² to a single 2-loop to obtain a 3-loop

module _
  {l : Level} {A : UU l} {a : A} (s : type-Ω² a)
  where

  3-loop-eckmann-hilton-Ω² : type-Ω³ a
  3-loop-eckmann-hilton-Ω² =
    map-pointed-equiv
      ( pointed-equiv-2-loop-pointed-identity (Ω (A , a)) (s ∙ s))
      ( eckmann-hilton-Ω² s s)

  inv-3-loop-eckmann-hilton-Ω² : type-Ω³ a
  inv-3-loop-eckmann-hilton-Ω² =
    map-pointed-equiv
      ( pointed-equiv-2-loop-pointed-identity (Ω (A , a)) (s ∙ s))
      ( inv-eckmann-hilton-Ω² s s)

The above two 3-loops are inverses

module _
  {l : Level} {A : UU l} {a : A} (s : type-Ω² a)
  where

  compute-inv-inv-3-loop-eckmann-hilton-Ω² :
    inv (inv-3-loop-eckmann-hilton-Ω² s) ＝ 3-loop-eckmann-hilton-Ω² s
  compute-inv-inv-3-loop-eckmann-hilton-Ω² =
    ( inv
      ( preserves-inv-map-Ω
        ( pointed-map-pointed-equiv
          ( pointed-equiv-loop-pointed-identity (Ω (A , a)) (s ∙ s)))
        (inv-eckmann-hilton-Ω² s s))) ∙
    ( ap
      ( map-Ω
        ( pointed-map-pointed-equiv
          ( pointed-equiv-loop-pointed-identity (Ω (A , a)) (s ∙ s))))
      ( compute-inv-inv-eckmann-hilton-Ω² s s))

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