π-finite types
Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Elisabeth Stenholm and Victor Blanchi.
Created on 2021-12-30.
Last modified on 2024-11-05.
module univalent-combinatorics.pi-finite-types where
Imports
open import elementary-number-theory.natural-numbers open import foundation.0-connected-types open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.dependent-identifications open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.empty-types open import foundation.equality-coproduct-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.fiber-inclusions open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-set-truncation open import foundation.homotopies open import foundation.identity-types open import foundation.logical-equivalences open import foundation.maybe open import foundation.mere-equality open import foundation.mere-equivalences open import foundation.propositional-extensionality open import foundation.propositional-truncations open import foundation.propositions open import foundation.set-presented-types open import foundation.set-truncations open import foundation.sets open import foundation.subtypes open import foundation.surjective-maps open import foundation.transport-along-identifications open import foundation.truncated-types open import foundation.truncation-levels open import foundation.type-arithmetic-coproduct-types open import foundation.unit-type open import foundation.univalence open import foundation.universe-levels open import foundation.whiskering-homotopies-composition open import univalent-combinatorics.coproduct-types open import univalent-combinatorics.counting open import univalent-combinatorics.dependent-function-types open import univalent-combinatorics.dependent-pair-types open import univalent-combinatorics.distributivity-of-set-truncation-over-finite-products open import univalent-combinatorics.equality-finite-types open import univalent-combinatorics.finite-types open import univalent-combinatorics.finitely-many-connected-components open import univalent-combinatorics.finitely-presented-types open import univalent-combinatorics.function-types open import univalent-combinatorics.image-of-maps open import univalent-combinatorics.standard-finite-types
Idea
A type is
πₙ-finite¶
if it has finitely many
connected components and all of its
homotopy groups up to level n at all base points are finite.
Definition
Truncated π-finite types
is-truncated-π-finite-Prop : {l : Level} (k : ℕ) → UU l → Prop l is-truncated-π-finite-Prop zero-ℕ X = is-finite-Prop X is-truncated-π-finite-Prop (succ-ℕ k) X = product-Prop ( has-finitely-many-connected-components-Prop X) ( Π-Prop X ( λ x → Π-Prop X (λ y → is-truncated-π-finite-Prop k (x ＝ y)))) is-truncated-π-finite : {l : Level} (k : ℕ) → UU l → UU l is-truncated-π-finite k A = type-Prop (is-truncated-π-finite-Prop k A)
π-finite types
is-π-finite-Prop : {l : Level} (k : ℕ) → UU l → Prop l is-π-finite-Prop zero-ℕ X = has-finitely-many-connected-components-Prop X is-π-finite-Prop (succ-ℕ k) X = product-Prop ( is-π-finite-Prop zero-ℕ X) ( Π-Prop X (λ x → Π-Prop X (λ y → is-π-finite-Prop k (x ＝ y)))) is-π-finite : {l : Level} (k : ℕ) → UU l → UU l is-π-finite k X = type-Prop (is-π-finite-Prop k X) is-prop-is-π-finite : {l : Level} (k : ℕ) (X : UU l) → is-prop (is-π-finite k X) is-prop-is-π-finite k X = is-prop-type-Prop (is-π-finite-Prop k X) π-Finite : (l : Level) (k : ℕ) → UU (lsuc l) π-Finite l k = Σ (UU l) (is-π-finite k) type-π-Finite : {l : Level} (k : ℕ) → π-Finite l k → UU l type-π-Finite k = pr1 is-π-finite-type-π-Finite : {l : Level} (k : ℕ) (A : π-Finite l k) → is-π-finite k (type-π-Finite {l} k A) is-π-finite-type-π-Finite k = pr2 has-finitely-many-connected-components-is-π-finite : {l : Level} (k : ℕ) {A : UU l} → is-π-finite k A → has-finitely-many-connected-components A has-finitely-many-connected-components-is-π-finite zero-ℕ H = H has-finitely-many-connected-components-is-π-finite (succ-ℕ k) H = pr1 H
Properties
π-finite types are closed under equivalences
is-π-finite-equiv : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} (e : A ≃ B) → is-π-finite k B → is-π-finite k A is-π-finite-equiv zero-ℕ e H = is-finite-equiv' (equiv-trunc-Set e) H pr1 (is-π-finite-equiv (succ-ℕ k) e H) = is-π-finite-equiv zero-ℕ e (pr1 H) pr2 (is-π-finite-equiv (succ-ℕ k) e H) a b = is-π-finite-equiv k ( equiv-ap e a b) ( pr2 H (map-equiv e a) (map-equiv e b)) is-π-finite-equiv' : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} (e : A ≃ B) → is-π-finite k A → is-π-finite k B is-π-finite-equiv' k e = is-π-finite-equiv k (inv-equiv e)
The empty type is π-finite
is-π-finite-empty : (k : ℕ) → is-π-finite k empty is-π-finite-empty zero-ℕ = is-finite-equiv equiv-unit-trunc-empty-Set is-finite-empty pr1 (is-π-finite-empty (succ-ℕ k)) = is-π-finite-empty zero-ℕ pr2 (is-π-finite-empty (succ-ℕ k)) = ind-empty empty-π-Finite : (k : ℕ) → π-Finite lzero k pr1 (empty-π-Finite k) = empty pr2 (empty-π-Finite k) = is-π-finite-empty k
Any empty type is π-finite
is-π-finite-is-empty : {l : Level} (k : ℕ) {A : UU l} → is-empty A → is-π-finite k A is-π-finite-is-empty zero-ℕ f = is-finite-is-empty (is-empty-trunc-Set f) pr1 (is-π-finite-is-empty (succ-ℕ k) f) = is-π-finite-is-empty zero-ℕ f pr2 (is-π-finite-is-empty (succ-ℕ k) f) a = ex-falso (f a)
Any contractible type is π-finite
is-π-finite-is-contr : {l : Level} (k : ℕ) {A : UU l} → is-contr A → is-π-finite k A is-π-finite-is-contr zero-ℕ H = is-finite-is-contr (is-contr-trunc-Set H) pr1 (is-π-finite-is-contr (succ-ℕ k) H) = is-π-finite-is-contr zero-ℕ H pr2 (is-π-finite-is-contr (succ-ℕ k) H) x y = is-π-finite-is-contr k ( is-prop-is-contr H x y)
The unit type is π-finite
is-π-finite-unit : (k : ℕ) → is-π-finite k unit is-π-finite-unit k = is-π-finite-is-contr k is-contr-unit unit-π-Finite : (k : ℕ) → π-Finite lzero k pr1 (unit-π-Finite k) = unit pr2 (unit-π-Finite k) = is-π-finite-unit k
Coproducts of π-finite types are π-finite
is-π-finite-coproduct : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} → is-π-finite k A → is-π-finite k B → is-π-finite k (A + B) is-π-finite-coproduct zero-ℕ H K = is-finite-equiv' ( equiv-distributive-trunc-coproduct-Set _ _) ( is-finite-coproduct H K) pr1 (is-π-finite-coproduct (succ-ℕ k) H K) = is-π-finite-coproduct zero-ℕ (pr1 H) (pr1 K) pr2 (is-π-finite-coproduct (succ-ℕ k) H K) (inl x) (inl y) = is-π-finite-equiv k ( compute-eq-coproduct-inl-inl x y) ( pr2 H x y) pr2 (is-π-finite-coproduct (succ-ℕ k) H K) (inl x) (inr y) = is-π-finite-equiv k ( compute-eq-coproduct-inl-inr x y) ( is-π-finite-empty k) pr2 (is-π-finite-coproduct (succ-ℕ k) H K) (inr x) (inl y) = is-π-finite-equiv k ( compute-eq-coproduct-inr-inl x y) ( is-π-finite-empty k) pr2 (is-π-finite-coproduct (succ-ℕ k) H K) (inr x) (inr y) = is-π-finite-equiv k ( compute-eq-coproduct-inr-inr x y) ( pr2 K x y) coproduct-π-Finite : {l1 l2 : Level} (k : ℕ) → π-Finite l1 k → π-Finite l2 k → π-Finite (l1 ⊔ l2) k pr1 (coproduct-π-Finite k A B) = (type-π-Finite k A + type-π-Finite k B) pr2 (coproduct-π-Finite k A B) = is-π-finite-coproduct k ( is-π-finite-type-π-Finite k A) ( is-π-finite-type-π-Finite k B)
Maybe A of any π-finite type A is π-finite
Maybe-π-Finite : {l : Level} (k : ℕ) → π-Finite l k → π-Finite l k Maybe-π-Finite k A = coproduct-π-Finite k A (unit-π-Finite k) is-π-finite-Maybe : {l : Level} (k : ℕ) {A : UU l} → is-π-finite k A → is-π-finite k (Maybe A) is-π-finite-Maybe k H = is-π-finite-coproduct k H (is-π-finite-unit k)
Any stanadard finite type is π-finite
is-π-finite-Fin : (k n : ℕ) → is-π-finite k (Fin n) is-π-finite-Fin k zero-ℕ = is-π-finite-empty k is-π-finite-Fin k (succ-ℕ n) = is-π-finite-Maybe k (is-π-finite-Fin k n) Fin-π-Finite : (k : ℕ) (n : ℕ) → π-Finite lzero k pr1 (Fin-π-Finite k n) = Fin n pr2 (Fin-π-Finite k n) = is-π-finite-Fin k n
Any type equipped with a counting is π-finite
is-π-finite-count : {l : Level} (k : ℕ) {A : UU l} → count A → is-π-finite k A is-π-finite-count k (n , e) = is-π-finite-equiv' k e (is-π-finite-Fin k n)
Any finite type is π-finite
is-π-finite-is-finite : {l : Level} (k : ℕ) {A : UU l} → is-finite A → is-π-finite k A is-π-finite-is-finite k {A} H = apply-universal-property-trunc-Prop H ( is-π-finite-Prop k A) ( is-π-finite-count k) π-finite-𝔽 : {l : Level} (k : ℕ) → 𝔽 l → π-Finite l k pr1 (π-finite-𝔽 k A) = type-𝔽 A pr2 (π-finite-𝔽 k A) = is-π-finite-is-finite k (is-finite-type-𝔽 A)
The type of all n-element types in UU l is π-finite
is-π-finite-UU-Fin : {l : Level} (k n : ℕ) → is-π-finite k (UU-Fin l n) is-π-finite-UU-Fin zero-ℕ n = has-finitely-many-connected-components-is-0-connected ( is-0-connected-UU-Fin n) pr1 (is-π-finite-UU-Fin (succ-ℕ k) n) = is-π-finite-UU-Fin zero-ℕ n pr2 (is-π-finite-UU-Fin (succ-ℕ k) n) x y = is-π-finite-equiv k ( equiv-equiv-eq-UU-Fin n x y) ( is-π-finite-is-finite k ( is-finite-≃ ( is-finite-has-finite-cardinality (n , (pr2 x))) ( is-finite-has-finite-cardinality (n , (pr2 y)))))
πₙ₊₁-finite types are πₙ-finite
is-π-finite-is-π-finite-succ-ℕ : {l : Level} (k : ℕ) {A : UU l} → is-π-finite (succ-ℕ k) A → is-π-finite k A is-π-finite-is-π-finite-succ-ℕ zero-ℕ H = has-finitely-many-connected-components-is-π-finite 1 H pr1 (is-π-finite-is-π-finite-succ-ℕ (succ-ℕ k) H) = has-finitely-many-connected-components-is-π-finite (succ-ℕ (succ-ℕ k)) H pr2 (is-π-finite-is-π-finite-succ-ℕ (succ-ℕ k) H) x y = is-π-finite-is-π-finite-succ-ℕ k (pr2 H x y)
πₙ₊₁-finite types are π₁-finite
is-π-finite-one-is-π-finite-succ-ℕ : {l : Level} (k : ℕ) {A : UU l} → is-π-finite (succ-ℕ k) A → is-π-finite 1 A is-π-finite-one-is-π-finite-succ-ℕ zero-ℕ H = H is-π-finite-one-is-π-finite-succ-ℕ (succ-ℕ k) H = is-π-finite-one-is-π-finite-succ-ℕ k ( is-π-finite-is-π-finite-succ-ℕ (succ-ℕ k) H)
πₙ-finite sets are finite
is-finite-is-π-finite : {l : Level} (k : ℕ) {A : UU l} → is-set A → is-π-finite k A → is-finite A is-finite-is-π-finite k H K = is-finite-equiv' ( equiv-unit-trunc-Set (_ , H)) ( has-finitely-many-connected-components-is-π-finite k K)
πₙ-finite n-truncated types are truncated πₙ-finite
is-truncated-π-finite-is-π-finite : {l : Level} (k : ℕ) {A : UU l} → is-trunc (truncation-level-ℕ k) A → is-π-finite k A → is-truncated-π-finite k A is-truncated-π-finite-is-π-finite zero-ℕ H K = is-finite-is-π-finite zero-ℕ H K pr1 (is-truncated-π-finite-is-π-finite (succ-ℕ k) H K) = pr1 K pr2 (is-truncated-π-finite-is-π-finite (succ-ℕ k) H K) x y = is-truncated-π-finite-is-π-finite k (H x y) (pr2 K x y)
truncated πₙ-finite types are πₙ-finite
is-π-finite-is-truncated-π-finite : {l : Level} (k : ℕ) {A : UU l} → is-truncated-π-finite k A → is-π-finite k A is-π-finite-is-truncated-π-finite zero-ℕ H = is-finite-equiv ( equiv-unit-trunc-Set (_ , (is-set-is-finite H))) ( H) pr1 (is-π-finite-is-truncated-π-finite (succ-ℕ k) H) = pr1 H pr2 (is-π-finite-is-truncated-π-finite (succ-ℕ k) H) x y = is-π-finite-is-truncated-π-finite k (pr2 H x y)
Finite products of π-finite types are π-finite
is-π-finite-Π : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} → is-finite A → ((a : A) → is-π-finite k (B a)) → is-π-finite k ((a : A) → B a) is-π-finite-Π zero-ℕ {A} {B} H K = is-finite-equiv' ( equiv-distributive-trunc-Π-is-finite-Set B H) ( is-finite-Π H K) pr1 (is-π-finite-Π (succ-ℕ k) H K) = is-π-finite-Π zero-ℕ H (λ a → pr1 (K a)) pr2 (is-π-finite-Π (succ-ℕ k) H K) f g = is-π-finite-equiv k ( equiv-funext) ( is-π-finite-Π k H (λ a → pr2 (K a) (f a) (g a))) π-Finite-Π : {l1 l2 : Level} (k : ℕ) (A : 𝔽 l1) (B : type-𝔽 A → π-Finite l2 k) → π-Finite (l1 ⊔ l2) k pr1 (π-Finite-Π k A B) = (x : type-𝔽 A) → (type-π-Finite k (B x)) pr2 (π-Finite-Π k A B) = is-π-finite-Π k ( is-finite-type-𝔽 A) ( λ x → is-π-finite-type-π-Finite k (B x))
Dependent sums of types with finitely many connected components over a 0-connected base
The total space of a family of types with finitely many connected components
over a 0-connected base has finitely many connected components when the based
loop spaces of the base have
finitely many connected components.
has-finitely-many-connected-components-Σ-is-0-connected : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → is-0-connected A → ((a : A) → has-finitely-many-connected-components (a ＝ a)) → ((x : A) → has-finitely-many-connected-components (B x)) → has-finitely-many-connected-components (Σ A B) has-finitely-many-connected-components-Σ-is-0-connected {A = A} {B} C H K = apply-universal-property-trunc-Prop ( is-inhabited-is-0-connected C) ( has-finitely-many-connected-components-Prop (Σ A B)) ( α) where α : A → has-finitely-many-connected-components (Σ A B) α a = is-finite-codomain ( K a) ( is-surjective-map-trunc-Set ( fiber-inclusion B a) ( is-surjective-fiber-inclusion C a)) ( apply-dependent-universal-property-trunc-Set' ( λ x → set-Prop ( Π-Prop ( type-trunc-Set (Σ A B)) ( λ y → is-decidable-Prop (Id-Prop (trunc-Set (Σ A B)) x y)))) ( β)) where β : (x : Σ A B) (v : type-trunc-Set (Σ A B)) → is-decidable (unit-trunc-Set x ＝ v) β (x , y) = apply-dependent-universal-property-trunc-Set' ( λ u → set-Prop ( is-decidable-Prop ( Id-Prop (trunc-Set (Σ A B)) (unit-trunc-Set (x , y)) u))) ( γ) where γ : (v : Σ A B) → is-decidable ((unit-trunc-Set (x , y)) ＝ (unit-trunc-Set v)) γ (x' , y') = is-decidable-equiv ( is-effective-unit-trunc-Set ( Σ A B) ( x , y) ( x' , y')) ( apply-universal-property-trunc-Prop ( mere-eq-is-0-connected C a x) ( is-decidable-Prop ( mere-eq-Prop (x , y) (x' , y'))) ( δ)) where δ : a ＝ x → is-decidable (mere-eq (x , y) (x' , y')) δ refl = apply-universal-property-trunc-Prop ( mere-eq-is-0-connected C a x') ( is-decidable-Prop ( mere-eq-Prop (a , y) (x' , y'))) ( ε) where ε : a ＝ x' → is-decidable (mere-eq (x , y) (x' , y')) ε refl = is-decidable-equiv e ( is-decidable-type-trunc-Prop-is-finite ( is-finite-Σ ( H a) ( λ ω → is-finite-is-decidable-Prop (P ω) (d ω)))) where ℙ : is-contr ( Σ ( hom-Set (trunc-Set (a ＝ a)) (Prop-Set _)) ( λ h → ( λ a₁ → h (unit-trunc-Set a₁)) ~ ( λ ω₁ → trunc-Prop (dependent-identification B ω₁ y y')))) ℙ = universal-property-trunc-Set ( a ＝ a) ( Prop-Set _) ( λ ω → trunc-Prop (dependent-identification B ω y y')) P : type-trunc-Set (Id a a) → Prop _ P = pr1 (center ℙ) compute-P : (ω : a ＝ a) → type-Prop (P (unit-trunc-Set ω)) ≃ type-trunc-Prop (dependent-identification B ω y y') compute-P ω = equiv-eq (ap pr1 (pr2 (center ℙ) ω)) d : (t : type-trunc-Set (a ＝ a)) → is-decidable (type-Prop (P t)) d = apply-dependent-universal-property-trunc-Set' ( λ t → set-Prop (is-decidable-Prop (P t))) ( λ ω → is-decidable-equiv' ( inv-equiv (compute-P ω)) ( is-decidable-equiv' ( is-effective-unit-trunc-Set (B a) (tr B ω y) y') ( has-decidable-equality-is-finite ( K a) ( unit-trunc-Set (tr B ω y)) ( unit-trunc-Set y')))) f : type-hom-Prop ( trunc-Prop (Σ (type-trunc-Set (Id a a)) (type-Prop ∘ P))) ( mere-eq-Prop {A = Σ A B} (a , y) (a , y')) f t = apply-universal-property-trunc-Prop t ( mere-eq-Prop (a , y) (a , y')) λ (u , v) → apply-dependent-universal-property-trunc-Set' ( λ u' → hom-set-Set ( set-Prop (P u')) ( set-Prop ( mere-eq-Prop (a , y) (a , y')))) ( λ ω v' → apply-universal-property-trunc-Prop ( map-equiv (compute-P ω) v') ( mere-eq-Prop (a , y) (a , y')) ( λ p → unit-trunc-Prop (eq-pair-Σ ω p))) ( u) ( v) e : mere-eq {A = Σ A B} (a , y) (a , y') ≃ type-trunc-Prop (Σ (type-trunc-Set (Id a a)) (type-Prop ∘ P)) e = equiv-iff ( mere-eq-Prop (a , y) (a , y')) ( trunc-Prop (Σ (type-trunc-Set (a ＝ a)) (type-Prop ∘ P))) ( λ t → apply-universal-property-trunc-Prop t ( trunc-Prop _) ( ( λ (ω , r) → unit-trunc-Prop ( ( unit-trunc-Set ω) , ( map-inv-equiv ( compute-P ω) ( unit-trunc-Prop r)))) ∘ ( pair-eq-Σ))) ( f)
Dependent sums of types with finitely many connected components
abstract has-finitely-many-connected-components-Σ' : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} → (Fin k ≃ type-trunc-Set A) → ((x y : A) → has-finitely-many-connected-components (x ＝ y)) → ((x : A) → has-finitely-many-connected-components (B x)) → has-finitely-many-connected-components (Σ A B) has-finitely-many-connected-components-Σ' zero-ℕ e H K = has-finitely-many-connected-components-is-empty ( is-empty-is-empty-trunc-Set (map-inv-equiv e) ∘ pr1) has-finitely-many-connected-components-Σ' (succ-ℕ k) {A} {B} e H K = apply-universal-property-trunc-Prop ( has-presentation-of-cardinality-has-cardinality-connected-components ( succ-ℕ k) ( unit-trunc-Prop e)) ( has-finitely-many-connected-components-Prop (Σ A B)) ( α) where α : Σ (Fin (succ-ℕ k) → A) (λ f → is-equiv (unit-trunc-Set ∘ f)) → has-finitely-many-connected-components (Σ A B) α (f , Eηf) = is-finite-equiv ( equiv-trunc-Set g) ( is-finite-equiv' ( equiv-distributive-trunc-coproduct-Set ( Σ (im (f ∘ inl)) (B ∘ pr1)) ( Σ (im (f ∘ inr)) (B ∘ pr1))) ( is-finite-coproduct ( has-finitely-many-connected-components-Σ' k ( h) ( λ x y → is-finite-equiv' ( equiv-trunc-Set ( equiv-ap-emb ( pr1 , is-emb-inclusion-subtype ( λ u → trunc-Prop _)))) ( H (pr1 x) (pr1 y))) ( λ x → K (pr1 x))) ( has-finitely-many-connected-components-Σ-is-0-connected ( is-0-connected-im-is-0-connected-domain ( f ∘ inr) ( is-0-connected-unit)) ( ( λ a → has-finitely-many-connected-components-equiv' ( equiv-Eq-eq-im (f ∘ inr) a a) ( H (pr1 a) (pr1 a)))) ( λ x → K (pr1 x))))) where g : ((Σ (im (f ∘ inl)) (B ∘ pr1)) + (Σ (im (f ∘ inr)) (B ∘ pr1))) ≃ Σ A B g = ( equiv-Σ B ( is-coproduct-codomain f ( unit-trunc-Set ∘ f , Eηf) ( refl-htpy)) ( λ { (inl x) → id-equiv ; (inr x) → id-equiv})) ∘e ( inv-equiv ( right-distributive-Σ-coproduct ( im (f ∘ inl)) ( im (f ∘ inr)) ( rec-coproduct (B ∘ pr1) (B ∘ pr1)))) i : Fin k → type-trunc-Set (im (f ∘ inl)) i = unit-trunc-Set ∘ map-unit-im (f ∘ inl) is-surjective-i : is-surjective i is-surjective-i = is-surjective-comp ( is-surjective-unit-trunc-Set (im (f ∘ inl))) ( is-surjective-map-unit-im (f ∘ inl)) is-emb-i : is-emb i is-emb-i = is-emb-top-map-triangle ( (unit-trunc-Set ∘ f) ∘ inl) ( inclusion-trunc-im-Set (f ∘ inl)) ( i) ( ( inv-htpy (naturality-unit-trunc-Set (inclusion-im (f ∘ inl)))) ·r ( map-unit-im (f ∘ inl))) ( is-emb-inclusion-trunc-im-Set (f ∘ inl)) ( is-emb-comp ( unit-trunc-Set ∘ f) ( inl) ( is-emb-is-equiv Eηf) ( is-emb-inl)) h : Fin k ≃ type-trunc-Set (im (f ∘ inl)) h = i , (is-equiv-is-emb-is-surjective is-surjective-i is-emb-i)
Dependent sums of π-finite types
The dependent sum of a family of πₙ-finite types over a πₙ₊₁-finite base is πₙ-finite.
has-finitely-many-connected-components-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → is-π-finite 1 A → ((x : A) → has-finitely-many-connected-components (B x)) → has-finitely-many-connected-components (Σ A B) has-finitely-many-connected-components-Σ {A = A} {B} H K = apply-universal-property-trunc-Prop ( pr1 H) ( has-finitely-many-connected-components-Prop (Σ A B)) ( λ (k , e) → has-finitely-many-connected-components-Σ' k e (λ x y → pr2 H x y) K) abstract is-π-finite-Σ : {l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} → is-π-finite (succ-ℕ k) A → ((x : A) → is-π-finite k (B x)) → is-π-finite k (Σ A B) is-π-finite-Σ zero-ℕ = has-finitely-many-connected-components-Σ pr1 (is-π-finite-Σ (succ-ℕ k) H K) = has-finitely-many-connected-components-Σ ( is-π-finite-one-is-π-finite-succ-ℕ (succ-ℕ k) H) ( λ x → has-finitely-many-connected-components-is-π-finite (succ-ℕ k) (K x)) pr2 (is-π-finite-Σ (succ-ℕ k) H K) (x , u) (y , v) = is-π-finite-equiv k ( equiv-pair-eq-Σ (x , u) (y , v)) ( is-π-finite-Σ k ( pr2 H x y) ( λ where refl → pr2 (K x) u v)) π-Finite-Σ : {l1 l2 : Level} (k : ℕ) (A : π-Finite l1 (succ-ℕ k)) (B : (x : type-π-Finite (succ-ℕ k) A) → π-Finite l2 k) → π-Finite (l1 ⊔ l2) k pr1 (π-Finite-Σ k A B) = Σ (type-π-Finite (succ-ℕ k) A) (λ x → type-π-Finite k (B x)) pr2 (π-Finite-Σ k A B) = is-π-finite-Σ k ( is-π-finite-type-π-Finite (succ-ℕ k) A) ( λ x → is-π-finite-type-π-Finite k (B x))
External links
- pi-finite type at Lab
Recent changes
- 2024-11-05. Fredrik Bakke. Some results about path-cosplit maps (#1167).
- 2024-08-22. Fredrik Bakke. Cleanup of finite types (#1166).
- 2024-04-11. Fredrik Bakke. Refactor diagonals (#1096).
- 2024-04-11. Fredrik Bakke and Egbert Rijke. Propositional operations (#1008).
- 2024-02-06. Fredrik Bakke. Rename
(co)prodto(co)product(#1017).