Simpson’s delooping of the sign homomorphism
Content created by Fredrik Bakke, Egbert Rijke, Eléonore Mangel, Jonathan Prieto-Cubides, Elisabeth Stenholm, Julian KG, fernabnor and louismntnu.
Created on 2022-07-07.
Last modified on 2024-03-12.
{-# OPTIONS --lossy-unification #-} module finite-group-theory.simpson-delooping-sign-homomorphism where
Imports
open import elementary-number-theory.addition-natural-numbers open import elementary-number-theory.congruence-natural-numbers open import elementary-number-theory.inequality-natural-numbers open import elementary-number-theory.modular-arithmetic-standard-finite-types open import elementary-number-theory.natural-numbers open import finite-group-theory.delooping-sign-homomorphism open import finite-group-theory.finite-type-groups open import finite-group-theory.permutations open import finite-group-theory.sign-homomorphism open import finite-group-theory.transpositions open import foundation.action-on-equivalences-type-families-over-subuniverses open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.coproduct-types open import foundation.decidable-equivalence-relations open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalence-classes open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences open import foundation.function-types open import foundation.identity-types open import foundation.involutions open import foundation.logical-equivalences open import foundation.mere-equivalences open import foundation.negation open import foundation.propositional-truncations open import foundation.raising-universe-levels open import foundation.sets open import foundation.transport-along-identifications open import foundation.type-theoretic-principle-of-choice open import foundation.unit-type open import foundation.universe-levels open import group-theory.concrete-groups open import group-theory.groups open import group-theory.homomorphisms-concrete-groups open import group-theory.homomorphisms-groups open import group-theory.isomorphisms-groups open import group-theory.loop-groups-sets open import group-theory.symmetric-groups open import lists.lists open import univalent-combinatorics.2-element-decidable-subtypes open import univalent-combinatorics.counting open import univalent-combinatorics.equality-finite-types open import univalent-combinatorics.equality-standard-finite-types open import univalent-combinatorics.finite-types open import univalent-combinatorics.standard-finite-types
Ideas
We give a definition of the delooping of the sign homomorphism based on a suggestion by Alex Simpson.
Definitions
module _ {l : Level} (n : ℕ) (X : UU-Fin l n) where sign-comp-equivalence-relation : equivalence-relation lzero (Fin n ≃ type-UU-Fin n X) pr1 sign-comp-equivalence-relation f g = Id-Prop ( Fin-Set 2) ( zero-Fin 1) ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv f ∘e g)) pr1 (pr2 sign-comp-equivalence-relation) f = ap pr1 { x = zero-Fin 1 , unit-trunc-Prop (nil , refl , left-inverse-law-equiv f)} { y = center ( is-contr-parity-transposition-permutation n (Fin-UU-Fin' n) (inv-equiv f ∘e f))} ( eq-is-contr ( is-contr-parity-transposition-permutation n (Fin-UU-Fin' n) (inv-equiv f ∘e f))) pr1 (pr2 (pr2 sign-comp-equivalence-relation)) f g P = ap pr1 { x = zero-Fin 1 , unit-trunc-Prop ( nil , refl , left-inverse-law-equiv (inv-equiv f ∘e g))} { y = center ( is-contr-parity-transposition-permutation n ( Fin-UU-Fin' n) ( inv-equiv (inv-equiv f ∘e g) ∘e (inv-equiv f ∘e g)))} ( eq-is-contr ( is-contr-parity-transposition-permutation n (Fin-UU-Fin' n) ( inv-equiv (inv-equiv f ∘e g) ∘e (inv-equiv f ∘e g)))) ∙ ( preserves-add-sign-homomorphism-Fin-two n ( Fin-UU-Fin' n) ( inv-equiv (inv-equiv f ∘e g)) ( inv-equiv f ∘e g) ∙ ( ap ( add-Fin 2 ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv (inv-equiv f ∘e g)))) ( inv P) ∙ ( ap ( mod-two-ℕ ∘ ( nat-Fin 2 ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv (inv-equiv f ∘e g)))) +ℕ_) ( is-zero-nat-zero-Fin {k = 1}) ∙ ( is-section-nat-Fin 1 ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv (inv-equiv f ∘e g))) ∙ ( ap ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n)) ( distributive-inv-comp-equiv g (inv-equiv f) ∙ ap (inv-equiv g ∘e_) (inv-inv-equiv f))))))) pr2 (pr2 (pr2 sign-comp-equivalence-relation)) f g h Q P = ( ap mod-two-ℕ ( ap ( zero-ℕ +ℕ_) ( inv (is-zero-nat-zero-Fin {k = 1}) ∙ ap (nat-Fin 2) Q) ∙ ( ap ( _+ℕ ( nat-Fin 2 ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv g ∘e h)))) ( inv (is-zero-nat-zero-Fin {k = 1}) ∙ ap (nat-Fin 2) P)))) ∙ ( inv ( preserves-add-sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv f ∘e g) (inv-equiv g ∘e h)) ∙ ( ap ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n)) ( associative-comp-equiv (inv-equiv g ∘e h) g (inv-equiv f) ∙ ( ap ( inv-equiv f ∘e_) ( inv (associative-comp-equiv h (inv-equiv g) g) ∙ ( ap (_∘e h) (right-inverse-law-equiv g) ∙ left-unit-law-equiv h)))))) is-decidable-sign-comp-equivalence-relation : (f g : Fin n ≃ type-UU-Fin n X) → is-decidable (sim-equivalence-relation sign-comp-equivalence-relation f g) is-decidable-sign-comp-equivalence-relation f g = has-decidable-equality-is-finite ( is-finite-Fin 2) ( zero-Fin 1) ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv f ∘e g)) quotient-sign-comp : UU (lsuc lzero ⊔ l) quotient-sign-comp = equivalence-class sign-comp-equivalence-relation quotient-sign-comp-Set : Set (lsuc lzero ⊔ l) quotient-sign-comp-Set = equivalence-class-Set sign-comp-equivalence-relation module _ {l : Level} {X : UU l} (eX : count X) (ineq : leq-ℕ 2 (number-of-elements-count eX)) where private transposition-eX : Fin (pr1 eX) ≃ Fin (pr1 eX) transposition-eX = transposition ( standard-2-Element-Decidable-Subtype ( has-decidable-equality-Fin (number-of-elements-count eX)) ( pr2 ( pr2 ( two-distinct-elements-leq-2-Fin ( number-of-elements-count eX) ( ineq))))) private abstract lemma : Id ( inr star) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv (equiv-count eX) ∘e (equiv-count eX ∘e transposition-eX))) lemma = ( inv ( eq-sign-homomorphism-Fin-two-transposition ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( standard-2-Element-Decidable-Subtype ( has-decidable-equality-Fin (number-of-elements-count eX)) ( pr2 ( pr2 ( two-distinct-elements-leq-2-Fin (number-of-elements-count eX) (ineq))))))) ∙ ( ap ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX))) ( inv (left-unit-law-equiv transposition-eX) ∙ ( ap ( _∘e transposition-eX) ( inv (left-inverse-law-equiv (equiv-count eX))) ∙ ( associative-comp-equiv ( transposition-eX) ( equiv-count eX) ( inv-equiv (equiv-count eX)))))) not-sign-comp-transposition-count : ( Y : 2-Element-Decidable-Subtype l X) → ¬ ( sim-equivalence-relation ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( transposition Y ∘e equiv-count eX) ( transposition Y ∘e (transposition Y ∘e equiv-count eX))) not-sign-comp-transposition-count Y P = neq-inl-inr ( P ∙ ( ap ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX))) ( ap ( inv-equiv (transposition Y ∘e equiv-count eX) ∘e_) ( inv ( associative-comp-equiv (equiv-count eX) (transposition Y) (transposition Y)) ∙ ( ap ( _∘e equiv-count eX) ( eq-htpy-equiv (is-involution-map-transposition Y)) ∙ ( left-unit-law-equiv (equiv-count eX)))) ∙ ( ap ( _∘e equiv-count eX) ( distributive-inv-comp-equiv (equiv-count eX) (transposition Y)) ∙ ( associative-comp-equiv ( equiv-count eX) ( inv-equiv (transposition Y)) ( inv-equiv (equiv-count eX)) ∙ ( ap ( λ h → inv-equiv (equiv-count eX) ∘e (h ∘e equiv-count eX)) ( own-inverse-is-involution ( is-involution-map-transposition Y)) ∙ ( ap ( λ h → inv-equiv (equiv-count eX) ∘e (transposition Y ∘e h)) ( inv (inv-inv-equiv (equiv-count eX)))))))) ∙ ( preserves-conjugation-sign-homomorphism-Fin-two ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX)) ( Fin-UU-Fin' (number-of-elements-count eX)) ( transposition Y) ( inv-equiv (equiv-count eX)) ∙ ( eq-sign-homomorphism-Fin-two-transposition ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX)) ( Y))))) inv-Fin-2-quotient-sign-comp-count : ( T : quotient-sign-comp ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) → is-decidable ( is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX)) → Fin 2 inv-Fin-2-quotient-sign-comp-count T (inl P) = inl (inr star) inv-Fin-2-quotient-sign-comp-count T (inr NP) = inr star equiv-Fin-2-quotient-sign-comp-count : Fin 2 ≃ quotient-sign-comp ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX)) pr1 equiv-Fin-2-quotient-sign-comp-count (inl (inr star)) = class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( equiv-count eX) pr1 equiv-Fin-2-quotient-sign-comp-count (inr star) = class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( equiv-count eX ∘e transposition-eX) pr2 equiv-Fin-2-quotient-sign-comp-count = is-equiv-is-invertible ( λ T → inv-Fin-2-quotient-sign-comp-count T ( is-decidable-is-in-equivalence-class-is-decidable ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( λ a b → has-decidable-equality-Fin 2 ( zero-Fin 1) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv a ∘e b))) ( T) ( equiv-count eX))) ( λ T → retraction-Fin-2-quotient-sign-comp-count T ( is-decidable-is-in-equivalence-class-is-decidable ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( λ a b → has-decidable-equality-Fin 2 ( zero-Fin 1) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv a ∘e b))) ( T) ( equiv-count eX))) ( λ k → section-Fin-2-quotient-sign-comp-count k ( is-decidable-is-in-equivalence-class-is-decidable ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( λ a b → has-decidable-equality-Fin 2 ( zero-Fin 1) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv a ∘e b))) ( pr1 equiv-Fin-2-quotient-sign-comp-count k) ( equiv-count eX))) where cases-retraction-Fin-2-quotient-sign-comp-count : ( T : quotient-sign-comp ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) → ¬ ( is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX)) → ( f : Fin (number-of-elements-count eX) ≃ X) → Id ( class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( f)) ( T) → ( k : Fin 2) → Id ( k) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv f ∘e equiv-count eX)) → is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX ∘e transposition-eX) cases-retraction-Fin-2-quotient-sign-comp-count T NP f p (inl (inr star)) q = ex-falso ( NP ( tr ( λ x → is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( x) ( equiv-count eX)) ( p) ( q))) cases-retraction-Fin-2-quotient-sign-comp-count T NP f p (inr star) q = tr ( λ x → is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( x) ( equiv-count eX ∘e transposition-eX)) ( p) ( eq-mod-succ-cong-ℕ 1 0 2 (cong-zero-ℕ' 2) ∙ ( ap-add-Fin 2 q lemma ∙ ( inv ( preserves-add-sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv f ∘e equiv-count eX) ( inv-equiv (equiv-count eX) ∘e ( equiv-count eX ∘e transposition-eX))) ∙ ( ap ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX))) ( associative-comp-equiv ( inv-equiv (equiv-count eX) ∘e ( equiv-count eX ∘e transposition-eX)) ( equiv-count eX) ( inv-equiv f) ∙ ( ap ( λ h → inv-equiv f ∘e (equiv-count eX ∘e h)) ( inv ( associative-comp-equiv ( transposition-eX) ( equiv-count eX) ( inv-equiv (equiv-count eX))) ∙ ( ap ( _∘e transposition-eX) ( left-inverse-law-equiv (equiv-count eX)) ∙ ( left-unit-law-equiv transposition-eX))))))))) retraction-Fin-2-quotient-sign-comp-count : ( T : quotient-sign-comp ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) → ( H : is-decidable ( is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX))) → Id ( pr1 equiv-Fin-2-quotient-sign-comp-count ( inv-Fin-2-quotient-sign-comp-count T H)) ( T) retraction-Fin-2-quotient-sign-comp-count T (inl P) = eq-effective-quotient' ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( equiv-count eX) ( T) ( P) retraction-Fin-2-quotient-sign-comp-count T (inr NP) = eq-effective-quotient' ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( equiv-count eX ∘e transposition-eX) ( T) ( apply-universal-property-trunc-Prop ( pr2 T) ( pair ( is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX ∘e transposition-eX)) ( is-prop-is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX ∘e transposition-eX))) ( λ (t , p) → cases-retraction-Fin-2-quotient-sign-comp-count T NP t ( inv ( eq-has-same-elements-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( t)) ( p))) ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX)) ( inv-equiv t ∘e equiv-count eX)) ( refl))) section-Fin-2-quotient-sign-comp-count : ( k : Fin 2) → ( D : is-decidable ( is-in-equivalence-class ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( pr1 equiv-Fin-2-quotient-sign-comp-count k) ( equiv-count eX))) → Id ( inv-Fin-2-quotient-sign-comp-count (pr1 equiv-Fin-2-quotient-sign-comp-count k) (D)) ( k) section-Fin-2-quotient-sign-comp-count (inl (inr star)) (inl D) = refl section-Fin-2-quotient-sign-comp-count (inl (inr star)) (inr ND) = ex-falso ( ND ( refl-equivalence-relation ( sign-comp-equivalence-relation ( number-of-elements-count eX) ( X , unit-trunc-Prop (equiv-count eX))) ( pr2 eX))) section-Fin-2-quotient-sign-comp-count (inr star) (inl D) = ex-falso ( neq-inr-inl ( lemma ∙ ( inv ( D ∙ ( ap ( sign-homomorphism-Fin-two ( number-of-elements-count eX) ( Fin-UU-Fin' (number-of-elements-count eX))) ( ap ( _∘e equiv-count eX) ( distributive-inv-comp-equiv ( transposition-eX) ( equiv-count eX)) ∙ ( associative-comp-equiv ( equiv-count eX) ( inv-equiv (equiv-count eX)) ( inv-equiv transposition-eX) ∙ ( ap ( inv-equiv transposition-eX ∘e_) ( left-inverse-law-equiv (equiv-count eX)) ∙ ( right-unit-law-equiv (inv-equiv transposition-eX) ∙ ( own-inverse-is-involution ( is-involution-map-transposition ( standard-2-Element-Decidable-Subtype ( has-decidable-equality-Fin ( number-of-elements-count eX)) ( pr2 ( pr2 ( two-distinct-elements-leq-2-Fin ( number-of-elements-count eX) ( ineq)))))) ∙ ( inv (left-unit-law-equiv transposition-eX) ∙ ( ap ( _∘e transposition-eX) ( inv ( left-inverse-law-equiv ( equiv-count eX))) ∙ ( associative-comp-equiv ( transposition-eX) ( equiv-count eX) ( inv-equiv (equiv-count eX))))))))))))))) section-Fin-2-quotient-sign-comp-count (inr star) (inr ND) = refl module _ {l : Level} (n : ℕ) (X : UU-Fin l n) (ineq : leq-ℕ 2 n) where equiv-fin-2-quotient-sign-comp-equiv-Fin : (Fin n ≃ type-UU-Fin n X) → (Fin 2 ≃ quotient-sign-comp n X) equiv-fin-2-quotient-sign-comp-equiv-Fin h = tr ( λ e → Fin 2 ≃ quotient-sign-comp n (type-UU-Fin n X , e)) ( all-elements-equal-type-trunc-Prop ( unit-trunc-Prop (equiv-count (n , h))) (pr2 X)) ( equiv-Fin-2-quotient-sign-comp-count (n , h) ineq)
module _ {l : Level} (n : ℕ) where map-simpson-comp-equiv : (X X' : UU-Fin l n) → (type-UU-Fin n X ≃ type-UU-Fin n X') → (Fin n ≃ type-UU-Fin n X) → (Fin n ≃ type-UU-Fin n X') map-simpson-comp-equiv X X' e f = e ∘e f simpson-comp-equiv : (X X' : UU-Fin l n) → (type-UU-Fin n X ≃ type-UU-Fin n X') → (Fin n ≃ type-UU-Fin n X) ≃ (Fin n ≃ type-UU-Fin n X') pr1 (simpson-comp-equiv X X' e) = map-simpson-comp-equiv X X' e pr2 (simpson-comp-equiv X X' e) = is-equiv-is-invertible ( map-simpson-comp-equiv X' X (inv-equiv e)) ( λ f → ( inv (associative-comp-equiv f (inv-equiv e) e)) ∙ ( ap (_∘e f) (right-inverse-law-equiv e) ∙ left-unit-law-equiv f)) ( λ f → ( inv (associative-comp-equiv f e (inv-equiv e))) ∙ ( ap (_∘e f) (left-inverse-law-equiv e) ∙ left-unit-law-equiv f)) abstract preserves-id-equiv-simpson-comp-equiv : (X : UU-Fin l n) → Id (simpson-comp-equiv X X id-equiv) id-equiv preserves-id-equiv-simpson-comp-equiv X = eq-htpy-equiv left-unit-law-equiv preserves-comp-simpson-comp-equiv : ( X Y Z : UU-Fin l n) ( e : type-UU-Fin n X ≃ type-UU-Fin n Y) → ( f : type-UU-Fin n Y ≃ type-UU-Fin n Z) → Id ( simpson-comp-equiv X Z (f ∘e e)) ( simpson-comp-equiv Y Z f ∘e simpson-comp-equiv X Y e) preserves-comp-simpson-comp-equiv X Y Z e f = eq-htpy-equiv ( λ h → associative-comp-equiv h e f) private lemma-sign-comp : ( X X' : UU-Fin l n) ( e : type-UU-Fin n X ≃ type-UU-Fin n X') → ( f f' : Fin n ≃ type-UU-Fin n X) → Id ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) (inv-equiv f ∘e f')) ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n) ( inv-equiv ( map-simpson-comp-equiv X X' e f) ∘e map-simpson-comp-equiv X X' e f')) lemma-sign-comp X X' e f f' = ap ( sign-homomorphism-Fin-two n (Fin-UU-Fin' n)) ( ap ( inv-equiv f ∘e_) ( inv (left-unit-law-equiv f') ∙ ( ap (_∘e f') (inv (left-inverse-law-equiv e)) ∙ ( associative-comp-equiv f' e (inv-equiv e)))) ∙ ( ( inv ( associative-comp-equiv (e ∘e f') (inv-equiv e) (inv-equiv f))) ∙ ( ap ( _∘e map-simpson-comp-equiv X X' e f') ( inv (distributive-inv-comp-equiv f e))))) preserves-sign-comp-simpson-comp-equiv : ( X X' : UU-Fin l n) ( e : type-UU-Fin n X ≃ type-UU-Fin n X') → ( f f' : Fin n ≃ type-UU-Fin n X) → ( sim-equivalence-relation (sign-comp-equivalence-relation n X) f f' ↔ sim-equivalence-relation ( sign-comp-equivalence-relation n X') ( map-simpson-comp-equiv X X' e f) ( map-simpson-comp-equiv X X' e f')) pr1 (preserves-sign-comp-simpson-comp-equiv X X' e f f') = _∙ lemma-sign-comp X X' e f f' pr2 (preserves-sign-comp-simpson-comp-equiv X X' e f f') = _∙ inv (lemma-sign-comp X X' e f f')
module _ {l : Level} where sign-comp-aut-succ-succ-Fin : (n : ℕ) → type-Group (symmetric-Group (raise-Fin-Set l (n +ℕ 2))) → Fin (n +ℕ 2) ≃ raise l (Fin (n +ℕ 2)) sign-comp-aut-succ-succ-Fin n = _∘e compute-raise l (Fin (n +ℕ 2)) not-action-equiv-family-on-subuniverse-transposition : ( n : ℕ) → ( Y : 2-Element-Decidable-Subtype l ( raise-Fin l (n +ℕ 2))) → ¬ ( sim-equivalence-relation ( sign-comp-equivalence-relation (n +ℕ 2) ( raise-Fin l (n +ℕ 2) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2)))) ( sign-comp-aut-succ-succ-Fin n (transposition Y)) ( map-equiv ( action-equiv-family-over-subuniverse ( mere-equiv-Prop (Fin (n +ℕ 2))) ( λ X → Fin (n +ℕ 2) ≃ pr1 X) ( raise l (Fin (n +ℕ 2)) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))) ( raise l (Fin (n +ℕ 2)) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))) ( transposition Y)) ( sign-comp-aut-succ-succ-Fin n (transposition Y)))) not-action-equiv-family-on-subuniverse-transposition n = tr ( λ f → ( Y : 2-Element-Decidable-Subtype l ( raise-Fin l (n +ℕ 2))) → ¬ ( sim-equivalence-relation ( sign-comp-equivalence-relation ( n +ℕ 2) ( raise-Fin l (n +ℕ 2) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2)))) ( sign-comp-aut-succ-succ-Fin n (transposition Y)) ( map-equiv ( f ( raise l (Fin (n +ℕ 2)) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))) ( raise l (Fin (n +ℕ 2)) , unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))) ( transposition Y)) ( sign-comp-aut-succ-succ-Fin n (transposition Y))))) ( ap pr1 { x = simpson-comp-equiv (n +ℕ 2) , preserves-id-equiv-simpson-comp-equiv (n +ℕ 2)} { y = ( action-equiv-family-over-subuniverse ( mere-equiv-Prop (Fin (n +ℕ 2))) ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X) , ( compute-id-equiv-action-equiv-family-over-subuniverse ( mere-equiv-Prop (Fin (n +ℕ 2))) ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)))} ( eq-is-contr ( is-contr-equiv' _ ( distributive-Π-Σ) ( is-contr-Π ( unique-action-equiv-family-over-subuniverse ( mere-equiv-Prop (Fin (n +ℕ 2))) ( λ Y → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) Y)))))) ( not-sign-comp-transposition-count (n +ℕ 2 , (compute-raise l (Fin (n +ℕ 2)))) (star)) simpson-delooping-sign : (n : ℕ) → hom-Concrete-Group (UU-Fin-Group l n) (UU-Fin-Group (lsuc lzero ⊔ l) 2) simpson-delooping-sign = quotient-delooping-sign ( λ n X → Fin n ≃ type-UU-Fin n X) ( sign-comp-equivalence-relation) ( λ n _ → is-decidable-sign-comp-equivalence-relation n) ( equiv-fin-2-quotient-sign-comp-equiv-Fin) ( sign-comp-aut-succ-succ-Fin) ( not-action-equiv-family-on-subuniverse-transposition) eq-simpson-delooping-sign-homomorphism : (n : ℕ) → Id ( comp-hom-Group ( symmetric-Group (raise-Fin-Set l (n +ℕ 2))) ( loop-group-Set (raise-Fin-Set l (n +ℕ 2))) ( group-Concrete-Group (UU-Fin-Group (lsuc lzero ⊔ l) 2)) ( comp-hom-Group ( loop-group-Set (raise-Fin-Set l (n +ℕ 2))) ( group-Concrete-Group (UU-Fin-Group l (n +ℕ 2))) ( group-Concrete-Group (UU-Fin-Group (lsuc lzero ⊔ l) 2)) ( hom-group-hom-Concrete-Group ( UU-Fin-Group l (n +ℕ 2)) ( UU-Fin-Group (lsuc lzero ⊔ l) 2) ( simpson-delooping-sign (n +ℕ 2))) ( hom-inv-iso-Group ( group-Concrete-Group (UU-Fin-Group l (n +ℕ 2))) ( loop-group-Set (raise-Fin-Set l (n +ℕ 2))) ( iso-loop-group-fin-UU-Fin-Group l (n +ℕ 2)))) ( hom-inv-symmetric-group-loop-group-Set (raise-Fin-Set l (n +ℕ 2)))) ( comp-hom-Group ( symmetric-Group (raise-Fin-Set l (n +ℕ 2))) ( symmetric-Group (Fin-Set (n +ℕ 2))) ( group-Concrete-Group (UU-Fin-Group (lsuc lzero ⊔ l) 2)) ( comp-hom-Group ( symmetric-Group (Fin-Set (n +ℕ 2))) ( symmetric-Group (Fin-Set 2)) ( group-Concrete-Group (UU-Fin-Group (lsuc lzero ⊔ l) 2)) ( symmetric-abstract-UU-fin-group-quotient-hom ( λ n X → Fin n ≃ type-UU-Fin n X) ( sign-comp-equivalence-relation) ( λ n H → is-decidable-sign-comp-equivalence-relation n) ( equiv-fin-2-quotient-sign-comp-equiv-Fin) ( sign-comp-aut-succ-succ-Fin) ( not-action-equiv-family-on-subuniverse-transposition) ( n)) ( sign-homomorphism ( n +ℕ 2) ( Fin (n +ℕ 2) , unit-trunc-Prop id-equiv))) ( hom-inv-symmetric-group-equiv-Set ( Fin-Set (n +ℕ 2)) ( raise-Fin-Set l (n +ℕ 2)) ( compute-raise l (Fin (n +ℕ 2))))) eq-simpson-delooping-sign-homomorphism = eq-quotient-delooping-sign-homomorphism ( λ n X → Fin n ≃ type-UU-Fin n X) ( sign-comp-equivalence-relation) ( λ n _ → is-decidable-sign-comp-equivalence-relation n) ( equiv-fin-2-quotient-sign-comp-equiv-Fin) ( sign-comp-aut-succ-succ-Fin) ( not-action-equiv-family-on-subuniverse-transposition)
References
- [MR23]
- Éléonore Mangel and Egbert Rijke. Delooping the sign homomorphism in univalent mathematics. 01 2023. arXiv:2301.10011.
Recent changes
- 2024-03-12. Fredrik Bakke. Bibliographies (#1058).
- 2023-11-24. Egbert Rijke. Abelianization (#877).
- 2023-11-04. Fredrik Bakke. Small fixes concrete groups (#897).
- 2023-09-12. Egbert Rijke. Beyond foundation (#751).
- 2023-09-11. Fredrik Bakke. Transport along and action on equivalences (#706).