Hatcher’s acyclic type
Content created by Fredrik Bakke, Egbert Rijke, Raymond Baker and Tom de Jong.
Created on 2023-04-26.
Last modified on 2024-06-03.
module synthetic-homotopy-theory.hatchers-acyclic-type where
Imports
open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.commuting-squares-of-identifications open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.path-algebra open import foundation.structure-identity-principle open import foundation.torsorial-type-families open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-dependent-pair-types open import foundation.universe-levels open import structured-types.pointed-maps open import structured-types.pointed-types open import structured-types.pointed-universal-property-contractible-types open import synthetic-homotopy-theory.acyclic-types open import synthetic-homotopy-theory.eckmann-hilton-argument open import synthetic-homotopy-theory.functoriality-loop-spaces open import synthetic-homotopy-theory.loop-spaces open import synthetic-homotopy-theory.powers-of-loops open import synthetic-homotopy-theory.suspensions-of-pointed-types open import synthetic-homotopy-theory.universal-property-suspensions-of-pointed-types
Idea
Hatcher’s acyclic type is a
higher inductive type equipped with a base point and
two loops a and b, and
identifications witnessing that a⁵ ＝ b³ and
b³ = (ab)². This type is acyclic, because the structure on Hatcher’s acyclic
type on any loop space is contractible.
Definitions
The structure of Hatcher’s acyclic type
structure-Hatcher-Acyclic-Type : {l : Level} → Pointed-Type l → UU l structure-Hatcher-Acyclic-Type A = Σ ( type-Ω A) ( λ a → Σ ( type-Ω A) ( λ b → ( power-nat-Ω 5 A a ＝ power-nat-Ω 3 A b) × ( power-nat-Ω 3 A b ＝ power-nat-Ω 2 A (a ∙ b))))
Algebras with the structure of Hatcher’s acyclic type
algebra-Hatcher-Acyclic-Type : (l : Level) → UU (lsuc l) algebra-Hatcher-Acyclic-Type l = Σ (Pointed-Type l) structure-Hatcher-Acyclic-Type
Morphisms of types equipped with the structure of Hatcher’s acyclic type
module _ {l1 l2 : Level} ((A , a1 , a2 , r1 , r2) : algebra-Hatcher-Acyclic-Type l1) ((B , b1 , b2 , s1 , s2) : algebra-Hatcher-Acyclic-Type l2) where is-hom-pointed-map-algebra-Hatcher-Acyclic-Type : (A →∗ B) → UU l2 is-hom-pointed-map-algebra-Hatcher-Acyclic-Type f = Σ ( map-Ω f a1 ＝ b1) ( λ u → Σ ( map-Ω f a2 ＝ b2) ( λ v → ( coherence-square-identifications ( ap (map-Ω f) r1) ( map-power-nat-Ω 5 f a1 ∙ ap (power-nat-Ω 5 B) u) ( map-power-nat-Ω 3 f a2 ∙ ap (power-nat-Ω 3 B) v) ( s1)) × ( coherence-square-identifications ( ap (map-Ω f) r2) ( map-power-nat-Ω 3 f a2 ∙ ap (power-nat-Ω 3 B) v) ( ( map-power-nat-Ω 2 f (a1 ∙ a2)) ∙ ( ap ( power-nat-Ω 2 B) ( ( preserves-mul-map-Ω f) ∙ ( horizontal-concat-Id² u v)))) ( s2)))) hom-algebra-Hatcher-Acyclic-Type : UU (l1 ⊔ l2) hom-algebra-Hatcher-Acyclic-Type = Σ ( A →∗ B) is-hom-pointed-map-algebra-Hatcher-Acyclic-Type
Initial Hatcher acyclic algebras
One characterization of Hatcher’s acyclic type is through its universal property as being the initial Hatcher acyclic algebra.
is-initial-algebra-Hatcher-Acyclic-Type : {l1 : Level} (A : algebra-Hatcher-Acyclic-Type l1) → UUω is-initial-algebra-Hatcher-Acyclic-Type A = {l : Level} → (B : algebra-Hatcher-Acyclic-Type l) → is-contr (hom-algebra-Hatcher-Acyclic-Type A B)
Properties
Characterization of identifications of Hatcher acyclic type structures
module _ {l : Level} (A : Pointed-Type l) (s : structure-Hatcher-Acyclic-Type A) where Eq-structure-Hatcher-Acyclic-Type : structure-Hatcher-Acyclic-Type A → UU l Eq-structure-Hatcher-Acyclic-Type t = Σ ( pr1 s ＝ pr1 t) ( λ p → Σ ( pr1 (pr2 s) ＝ pr1 (pr2 t)) ( λ q → ( ( pr1 (pr2 (pr2 s)) ∙ ap (power-nat-Ω 3 A) q) ＝ ( (ap (power-nat-Ω 5 A) p) ∙ pr1 (pr2 (pr2 t)))) × ( ( pr2 (pr2 (pr2 s)) ∙ ap (power-nat-Ω 2 A) (horizontal-concat-Id² p q)) ＝ ( ap (power-nat-Ω 3 A) q ∙ pr2 (pr2 (pr2 t)))))) refl-Eq-structure-Hatcher-Acyclic-Type : Eq-structure-Hatcher-Acyclic-Type s pr1 refl-Eq-structure-Hatcher-Acyclic-Type = refl pr1 (pr2 refl-Eq-structure-Hatcher-Acyclic-Type) = refl pr1 (pr2 (pr2 refl-Eq-structure-Hatcher-Acyclic-Type)) = right-unit pr2 (pr2 (pr2 refl-Eq-structure-Hatcher-Acyclic-Type)) = right-unit is-torsorial-Eq-structure-Hatcher-Acyclic-Type : is-torsorial Eq-structure-Hatcher-Acyclic-Type is-torsorial-Eq-structure-Hatcher-Acyclic-Type = is-torsorial-Eq-structure ( is-torsorial-Id (pr1 s)) ( pr1 s , refl) ( is-torsorial-Eq-structure ( is-torsorial-Id (pr1 (pr2 s))) ( pr1 (pr2 s) , refl) ( is-torsorial-Eq-structure ( is-torsorial-Id (pr1 (pr2 (pr2 s)) ∙ refl)) ( pr1 (pr2 (pr2 s)) , right-unit) ( is-torsorial-Id (pr2 (pr2 (pr2 s)) ∙ refl)))) Eq-eq-structure-Hatcher-Acyclic-Type : (t : structure-Hatcher-Acyclic-Type A) → (s ＝ t) → Eq-structure-Hatcher-Acyclic-Type t Eq-eq-structure-Hatcher-Acyclic-Type .s refl = refl-Eq-structure-Hatcher-Acyclic-Type is-equiv-Eq-eq-structure-Hatcher-Acyclic-Type : (t : structure-Hatcher-Acyclic-Type A) → is-equiv (Eq-eq-structure-Hatcher-Acyclic-Type t) is-equiv-Eq-eq-structure-Hatcher-Acyclic-Type = fundamental-theorem-id is-torsorial-Eq-structure-Hatcher-Acyclic-Type Eq-eq-structure-Hatcher-Acyclic-Type extensionality-structure-Hatcher-Acyclic-Type : (t : structure-Hatcher-Acyclic-Type A) → (s ＝ t) ≃ Eq-structure-Hatcher-Acyclic-Type t pr1 (extensionality-structure-Hatcher-Acyclic-Type t) = Eq-eq-structure-Hatcher-Acyclic-Type t pr2 (extensionality-structure-Hatcher-Acyclic-Type t) = is-equiv-Eq-eq-structure-Hatcher-Acyclic-Type t eq-Eq-structure-Hatcher-Acyclic-Type : (t : structure-Hatcher-Acyclic-Type A) → Eq-structure-Hatcher-Acyclic-Type t → s ＝ t eq-Eq-structure-Hatcher-Acyclic-Type t = map-inv-equiv (extensionality-structure-Hatcher-Acyclic-Type t)
Loop spaces uniquely have the structure of a Hatcher acyclic type
module _ {l : Level} (A : Pointed-Type l) where is-contr-structure-Hatcher-Acyclic-Type-Ω : is-contr (structure-Hatcher-Acyclic-Type (Ω A)) is-contr-structure-Hatcher-Acyclic-Type-Ω = is-contr-equiv ( Σ (type-Ω (Ω A)) (λ a → refl ＝ a)) ( equiv-tot ( λ a → ( ( inv-equiv ( equiv-ap ( equiv-concat' (refl-Ω A) (power-nat-Ω 5 (Ω A) a)) ( refl-Ω (Ω A)) ( a))) ∘e ( equiv-concat' ( power-nat-Ω 5 (Ω A) a) ( ( inv (power-nat-mul-Ω 3 2 (Ω A) a)) ∙ ( power-nat-succ-Ω' 5 (Ω A) a)))) ∘e ( ( ( left-unit-law-Σ-is-contr ( is-torsorial-Id' (a ∙ a)) ( a ∙ a , refl)) ∘e ( inv-associative-Σ ( type-Ω (Ω A)) ( λ b → b ＝ (a ∙ a)) ( λ bq → power-nat-Ω 5 (Ω A) a ＝ power-nat-Ω 3 (Ω A) (pr1 bq)))) ∘e ( equiv-tot ( λ b → ( commutative-product) ∘e ( equiv-product ( id-equiv) ( ( ( inv-equiv ( equiv-ap ( equiv-concat' (refl-Ω A) (b ∙ b)) ( b) ( a ∙ a))) ∘e ( equiv-concat ( inv (power-nat-add-Ω 1 2 (Ω A) b)) ( (a ∙ a) ∙ (b ∙ b)))) ∘e ( equiv-concat' ( power-nat-Ω 3 (Ω A) b) ( interchange-concat-Ω² a b a b))))))))) ( is-torsorial-Id refl)
For a fixed pointed map, the is-hom-pointed-map-algebra-Hatcher-Acyclic-Type family is torsorial
In proving this, it is helpful to consider an equivalent formulation of
is-hom-pointed-map-algebra-Hatcher-Acyclic-Type for which
torsoriality is almost immediate.
module _ {l1 l2 : Level} ((A , a1 , a2 , r1 , r2) : algebra-Hatcher-Acyclic-Type l1) ((B , b1 , b2 , s1 , s2) : algebra-Hatcher-Acyclic-Type l2) where is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' : (A →∗ B) → UU l2 is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' f = Σ ( map-Ω f a1 ＝ b1) ( λ u → Σ ( map-Ω f a2 ＝ b2) ( λ v → ( Id ( inv (map-power-nat-Ω 5 f a1 ∙ ap (power-nat-Ω 5 B) u) ∙ ( ap (map-Ω f) r1 ∙ (map-power-nat-Ω 3 f a2 ∙ ap (power-nat-Ω 3 B) v))) ( s1)) × ( Id ( inv (map-power-nat-Ω 3 f a2 ∙ ap (power-nat-Ω 3 B) v) ∙ ( ap (map-Ω f) r2 ∙ ( ( map-power-nat-Ω 2 f (a1 ∙ a2)) ∙ ( ap ( power-nat-Ω 2 B) ( ( preserves-mul-map-Ω f) ∙ ( horizontal-concat-Id² u v)))))) ( s2)))) module _ {l1 l2 : Level} ((A , σ) : algebra-Hatcher-Acyclic-Type l1) ((B , τ) : algebra-Hatcher-Acyclic-Type l2) where equiv-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type : (f : A →∗ B) → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type (A , σ) (B , τ) f ≃ is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' (A , σ) (B , τ) f equiv-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type f = equiv-tot ( λ p → equiv-tot ( λ q → equiv-product ( equiv-left-transpose-eq-concat' _ _ _ ∘e equiv-inv _ _) ( equiv-left-transpose-eq-concat' _ _ _ ∘e equiv-inv _ _))) module _ {l1 l2 : Level} (A : Pointed-Type l1) (B : Pointed-Type l2) (σ@(a1 , a2 , r1 , r2) : structure-Hatcher-Acyclic-Type A) (f : A →∗ B) where is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' : is-torsorial ( λ τ → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' (A , σ) (B , τ) f) is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' = is-torsorial-Eq-structure ( is-torsorial-Id (map-Ω f a1)) ((map-Ω f a1) , refl) ( is-torsorial-Eq-structure ( is-torsorial-Id (map-Ω f a2)) ((map-Ω f a2) , refl) ( is-torsorial-Eq-structure ( is-torsorial-Id _) ( _ , refl) ( is-torsorial-Id _))) abstract is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type : is-torsorial ( λ τ → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type (A , σ) (B , τ) f) is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type = is-contr-equiv ( Σ ( structure-Hatcher-Acyclic-Type B) ( λ τ → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' ( A , σ) ( B , τ) ( f))) ( equiv-tot ( λ τ → equiv-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type ( A , σ) ( B , τ) ( f))) ( is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type')
Characterization of pointed maps out of Hatcher’s acyclic type
For the initial Hatcher acyclic algebra A and any pointed type B, we exhibit
an equivalence (A →∗ B) ≃ structure-Hatcher-Acyclic-Type B.
We first show that for any Hatcher acyclic algebra A (so not necessarily the
initial one) and a pointed type B, a pointed map f : A →∗ B induces a
Hatcher acyclic structure on B. Moreover, with this induced structure, the map
f becomes a homomorphism of Hatcher acyclic algebras.
module _ {l1 l2 : Level} (A : Pointed-Type l1) (B : Pointed-Type l2) (σ@(a1 , a2 , r1 , r2) : structure-Hatcher-Acyclic-Type A) where map-Ω-structure-Hatcher-Acyclic-Type : (f : A →∗ B) → ( power-nat-Ω 5 B (map-Ω f a1) ＝ power-nat-Ω 3 B (map-Ω f a2)) × ( power-nat-Ω 3 B (map-Ω f a2) ＝ power-nat-Ω 2 B (mul-Ω B (map-Ω f a1) (map-Ω f a2))) pr1 (map-Ω-structure-Hatcher-Acyclic-Type f) = inv (map-power-nat-Ω 5 f a1) ∙ (ap (map-Ω f) r1 ∙ map-power-nat-Ω 3 f a2) pr2 (map-Ω-structure-Hatcher-Acyclic-Type f) = inv (map-power-nat-Ω 3 f a2) ∙ ( ap (map-Ω f) r2 ∙ ( map-power-nat-Ω 2 f (a1 ∙ a2) ∙ ap (power-nat-Ω 2 B) (preserves-mul-map-Ω f))) structure-Hatcher-Acyclic-Type-pointed-map : (A →∗ B) → structure-Hatcher-Acyclic-Type B pr1 (structure-Hatcher-Acyclic-Type-pointed-map f) = map-Ω f a1 pr1 (pr2 (structure-Hatcher-Acyclic-Type-pointed-map f)) = map-Ω f a2 (pr2 (pr2 (structure-Hatcher-Acyclic-Type-pointed-map f))) = map-Ω-structure-Hatcher-Acyclic-Type f hom-algebra-Hatcher-Acyclic-Type-pointed-map' : (f : A →∗ B) → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type' ( A , σ) ( B , structure-Hatcher-Acyclic-Type-pointed-map f) ( f) pr1 (hom-algebra-Hatcher-Acyclic-Type-pointed-map' f) = refl pr1 (pr2 (hom-algebra-Hatcher-Acyclic-Type-pointed-map' f)) = refl pr1 (pr2 (pr2 (hom-algebra-Hatcher-Acyclic-Type-pointed-map' f))) = ap-binary ( λ p q → inv p ∙ (ap (map-Ω f) r1 ∙ q)) ( right-unit {p = map-power-nat-Ω 5 f a1}) ( right-unit) pr2 (pr2 (pr2 (hom-algebra-Hatcher-Acyclic-Type-pointed-map' f))) = ap-binary ( λ p q → inv p ∙ ( ap (map-Ω f) r2 ∙ ( map-power-nat-Ω 2 f (a1 ∙ a2) ∙ ap (power-nat-Ω 2 B) q))) ( right-unit {p = map-power-nat-Ω 3 f a2}) ( right-unit {p = preserves-mul-map-Ω f}) hom-algebra-Hatcher-Acyclic-Type-pointed-map : (f : A →∗ B) → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type ( A , σ) ( B , structure-Hatcher-Acyclic-Type-pointed-map f) ( f) hom-algebra-Hatcher-Acyclic-Type-pointed-map f = map-inv-equiv ( equiv-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type ( A , σ) ( B , structure-Hatcher-Acyclic-Type-pointed-map f) ( f)) ( hom-algebra-Hatcher-Acyclic-Type-pointed-map' f) is-equiv-structure-Hatcher-Acyclic-Type-pointed-map : is-initial-algebra-Hatcher-Acyclic-Type (A , σ) → is-equiv structure-Hatcher-Acyclic-Type-pointed-map is-equiv-structure-Hatcher-Acyclic-Type-pointed-map i = is-equiv-htpy-equiv ( equivalence-reasoning A →∗ B ≃ Σ ( A →∗ B) ( λ f → Σ ( structure-Hatcher-Acyclic-Type B) ( λ τ → is-hom-pointed-map-algebra-Hatcher-Acyclic-Type ( A , σ) ( B , τ) ( f))) by inv-right-unit-law-Σ-is-contr ( λ f → is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type ( A) ( B) ( σ) ( f)) ≃ Σ ( structure-Hatcher-Acyclic-Type B) ( λ τ → hom-algebra-Hatcher-Acyclic-Type (A , σ) (B , τ)) by equiv-left-swap-Σ ≃ structure-Hatcher-Acyclic-Type B by equiv-pr1 (λ τ → i (B , τ))) ( λ f → ap ( pr1) ( inv ( contraction ( is-torsorial-is-hom-pointed-map-algebra-Hatcher-Acyclic-Type A B ( σ) ( f)) ( structure-Hatcher-Acyclic-Type-pointed-map f , hom-algebra-Hatcher-Acyclic-Type-pointed-map f)))) equiv-pointed-map-Hatcher-Acyclic-Type : is-initial-algebra-Hatcher-Acyclic-Type (A , σ) → (A →∗ B) ≃ structure-Hatcher-Acyclic-Type B pr1 (equiv-pointed-map-Hatcher-Acyclic-Type i) = structure-Hatcher-Acyclic-Type-pointed-map pr2 (equiv-pointed-map-Hatcher-Acyclic-Type i) = is-equiv-structure-Hatcher-Acyclic-Type-pointed-map i
Hatcher’s acyclic type is acyclic
Proof: For the Hatcher acyclic type A, and any pointed type X, we have
(Σ A →∗ X) ≃ (A →∗ Ω X) ≃ structure-Hatcher-Acyclic-Type (Ω X),
where the first equivalence is the
suspension-loop space adjunction,
the second equivalence was just proven above, and the latter type is
contractible, as shown by is-contr-structure-Hatcher-Acyclic-Type-Ω.
Hence, the suspension Σ A of A satisfies the
universal property of contractible types with respect to pointed types and maps,
and hence must be contractible.
module _ {l1 : Level} ((A , σ) : algebra-Hatcher-Acyclic-Type l1) where is-acyclic-is-initial-algebra-Hatcher-Acyclic-Type : is-initial-algebra-Hatcher-Acyclic-Type (A , σ) → is-acyclic (type-Pointed-Type A) is-acyclic-is-initial-algebra-Hatcher-Acyclic-Type i = is-contr-universal-property-contr-Pointed-Type ( suspension-Pointed-Type A) ( λ X → is-contr-equiv ( structure-Hatcher-Acyclic-Type (Ω X)) ( ( equiv-pointed-map-Hatcher-Acyclic-Type A (Ω X) σ i) ∘e ( equiv-transpose-suspension-loop-adjunction A X)) ( is-contr-structure-Hatcher-Acyclic-Type-Ω X))
See also
Table of files related to cyclic types, groups, and rings
Recent changes
- 2024-06-03. Egbert Rijke. Change equiv-comp to comp-equiv (#1151).
- 2024-02-06. Fredrik Bakke. Rename
(co)prodto(co)product(#1017). - 2024-02-06. Egbert Rijke and Fredrik Bakke. Refactor files about identity types and homotopies (#1014).
- 2024-01-31. Fredrik Bakke. Rename
is-torsorial-pathtois-torsorial-Id(#1016). - 2024-01-17. Tom de Jong. Hatcher’s acyclic type is acyclic (#1003).