module Homotopy.Connectedness where

# Connectedness🔗

We say that a type is
**$n-connected$**
if its
$n-$truncation is contractible.

While being
$n-$truncated expresses that all
homotopy groups above
$n$
are trivial, being
$n-connected$
means that all homotopy groups *below*
$n$
are trivial. A type that is both
$n-truncated$
and
$n-connected$
is contractible.

We give definitions in terms of the propositional truncation and set truncation for the lower levels, and then defer to the general “hubs and spokes” truncation. Note that our indices are offset by 2, just like h-levels.

is-n-connected : ∀ {ℓ} → Type ℓ → Nat → Type _ is-n-connected A zero = Lift _ ⊤ is-n-connected A (suc zero) = ∥ A ∥ is-n-connected A (suc (suc zero)) = is-contr ∥ A ∥₀ is-n-connected A n@(suc (suc (suc _))) = is-contr (n-Tr A n)

Being $n-connected$ is a proposition:

is-n-connected-is-prop : (n : Nat) → is-prop (is-n-connected A n) is-n-connected-is-prop 0 _ _ = refl is-n-connected-is-prop 1 = is-prop-∥-∥ is-n-connected-is-prop 2 = is-contr-is-prop is-n-connected-is-prop (suc (suc (suc n))) = is-contr-is-prop

For short, we say that a type is **connected** if it is
$0-connected,$
and **simply connected** if it is
$1-connected:$

is-connected : ∀ {ℓ} → Type ℓ → Type _ is-connected A = is-n-connected A 2 is-simply-connected : ∀ {ℓ} → Type ℓ → Type _ is-simply-connected A = is-n-connected A 3

Furthermore, we say that a map is $n-connected$ if all of its fibres are $n-connected.$

is-n-connected-map : (A → B) → Nat → Type _ is-n-connected-map f n = ∀ x → is-n-connected (fibre f x) n

## Pointed connected types🔗

In the case of pointed types, there is an equivalent definition of being connected that is sometimes easier to work with: a pointed type is connected if every point is merely equal to the base point.

is-connected∙ : ∀ {ℓ} → Type∙ ℓ → Type _ is-connected∙ (X , pt) = (x : X) → ∥ x ≡ pt ∥ module _ {ℓ} {X@(_ , pt) : Type∙ ℓ} where is-connected∙→is-connected : is-connected∙ X → is-connected ⌞ X ⌟ is-connected∙→is-connected c .centre = inc pt is-connected∙→is-connected c .paths = elim! λ x → case c x of λ pt=x → ap inc (sym pt=x) is-connected→is-connected∙ : is-connected ⌞ X ⌟ → is-connected∙ X is-connected→is-connected∙ c x = ∥-∥₀-path.to (is-contr→is-prop c (inc x) (inc pt))

This alternative definition lets us formulate a nice elimination principle for pointed connected types: any family of propositions $P$ that holds on the base point holds everywhere.

In particular, since `being a proposition is a proposition`

,
we only need to check that
$P(∙)$
is a proposition.

connected∙-elim-prop : ∀ {ℓ ℓ'} {X : Type∙ ℓ} {P : ⌞ X ⌟ → Type ℓ'} → is-connected∙ X → is-prop (P (X .snd)) → P (X .snd) → ∀ x → P x connected∙-elim-prop {X = X} {P} conn prop pb x = ∥-∥-rec propx (λ e → subst P (sym e) pb) (conn x) where abstract propx : is-prop (P x) propx = ∥-∥-rec is-prop-is-prop (λ e → subst (is-prop ∘ P) (sym e) prop) (conn x)

We can similarly define an elimination principle into sets.

module connected∙-elim-set {ℓ ℓ'} {X : Type∙ ℓ} {P : ⌞ X ⌟ → Type ℓ'} (conn : is-connected∙ X) (set : is-set (P (X .snd))) (pb : P (X .snd)) (loops : ∀ (p : X .snd ≡ X .snd) → PathP (λ i → P (p i)) pb pb) where opaque elim : ∀ x → P x elim x = work (conn x) module elim where setx : is-set (P x) setx = ∥-∥-rec (is-hlevel-is-prop 2) (λ e → subst (is-set ∘ P) (sym e) set) (conn x) work : ∥ x ≡ X .snd ∥ → P x work = ∥-∥-rec-set setx (λ p → subst P (sym p) pb) (λ p q i → subst P (sym (∙-filler'' (sym p) q i)) (loops (sym p ∙ q) i)) elim-β-point : elim (X .snd) ≡ pb elim-β-point = subst (λ c → elim.work (X .snd) c ≡ pb) (squash (inc refl) (conn (X .snd))) (transport-refl pb)

Examples of pointed connected types include the circle and the delooping of a group.

## Closure properties🔗

As a property of types, $n-connectedness$ enjoys closure properties very similar to those of $n-truncatedness;$ and we establish them in much the same way, by proving that $n-connectedness$ is preserved by retractions.

However, the definition of `is-n-connected`

, which uses the
explicit constructions of truncations for the base cases, is slightly
annoying to work when
$n$
is arbitrary: that’s the trade-off for it being easy to work with when
$n$
is a known, and usually small, number. Therefore, we prove the following
lemma by the recursion, establishing that the notion of connectedness
could very well have been defined using the general
$n-truncation$
uniformly.

is-n-connected-Tr : ∀ {ℓ} {A : Type ℓ} n → is-n-connected A (suc n) → is-contr (n-Tr A (suc n)) is-n-connected-Tr zero a-conn = ∥-∥-out! do pt ← a-conn pure $ contr (inc pt) (λ x → n-Tr-is-hlevel 0 _ _) is-n-connected-Tr (suc zero) a-conn = retract→is-hlevel 0 (∥-∥₀-rec (n-Tr-is-hlevel 1) inc) (n-Tr-rec squash inc) (n-Tr-elim _ (λ _ → is-prop→is-set (n-Tr-is-hlevel 1 _ _)) λ _ → refl) a-conn is-n-connected-Tr (suc (suc n)) a-conn = a-conn

is-n-connected-Tr-is-equiv : ∀ {ℓ} {A : Type ℓ} n → is-equiv (is-n-connected-Tr {A = A} n) is-n-connected-Tr-is-equiv {A = A} n = prop-ext (is-n-connected-is-prop _) (hlevel 1) _ (from n) .snd where from : ∀ n → is-contr (n-Tr A (suc n)) → is-n-connected A (suc n) from zero c = n-Tr-rec! inc (c .centre) from (suc zero) = retract→is-hlevel 0 (n-Tr-rec! inc) (∥-∥₀-rec (n-Tr-is-hlevel 1) inc) (∥-∥₀-elim (λ _ → is-prop→is-set (squash _ _)) λ _ → refl) from (suc (suc n)) x = x module n-connected {ℓ} {A : Type ℓ} (n : Nat) = Equiv (_ , is-n-connected-Tr-is-equiv {A = A} n)

The first closure property we’ll prove is very applicable: it says that any retract of an $n-connected$ type is again $n-connected.$ Intuitively this is because any retraction $f:A→B$ can be extended to a retraction $∥A∥_{n}→∥B∥_{n},$ and contractible types are closed under retractions.

retract→is-n-connected : (n : Nat) (f : A → B) (g : B → A) → is-left-inverse f g → is-n-connected A n → is-n-connected B n retract→is-n-connected 0 = _ retract→is-n-connected 1 f g h a-conn = f <$> a-conn retract→is-n-connected (suc (suc n)) f g h a-conn = n-connected.from (suc n) $ retract→is-contr (rec! (inc ∘ f)) (rec! (inc ∘ g)) (elim! λ x → ap n-Tr.inc (h x)) (is-n-connected-Tr (suc n) a-conn)

Since the truncation operator $∥−∥_{n}$ also preserves products, a remarkably similar argument shows that if $A$ and $B$ are $n-connected,$ then so is $A×B$ is.

×-is-n-connected : ∀ n → is-n-connected A n → is-n-connected B n → is-n-connected (A × B) n ×-is-n-connected 0 = _ ×-is-n-connected (suc n) a-conn b-conn = n-connected.from n $ Equiv→is-hlevel 0 n-Tr-product (×-is-hlevel 0 (n-connected.to n a-conn) (n-connected.to n b-conn))

Finally, we show the dual of two properties of truncations: if
$A$
is
$(1+n)-connected,$
then each path space
$x≡_{A}y$
is
$n-connected;$
And if
$A$
is
$(2+n)-connected,$
then it is also
$(1+n)$
connected. This latter property lets us count *down* in precisely
the same way that `is-hlevel-suc`

lets us count
*up*.

Path-is-connected : ∀ n → is-n-connected A (suc n) → is-n-connected (Path A x y) n Path-is-connected 0 = _ Path-is-connected {x = x} (suc n) conn = n-connected.from n (contr (ps _ _) $ n-Tr-elim! _ (J (λ y p → ps x y ≡ inc p) (Equiv.injective (Equiv.inverse n-Tr-path-equiv) (is-contr→is-set (is-n-connected-Tr _ conn) _ _ _ _)))) where ps : ∀ x y → n-Tr (x ≡ y) (suc n) ps x y = Equiv.to n-Tr-path-equiv (is-contr→is-prop (is-n-connected-Tr _ conn) _ _) is-connected-suc : ∀ {ℓ} {A : Type ℓ} n → is-n-connected A (suc n) → is-n-connected A n is-connected-suc {A = A} zero _ = _ is-connected-suc {A = A} (suc n) w = n-connected.from n $ elim! (λ x → contr (inc x) (elim! (rem₁ n w x))) (is-n-connected-Tr (suc n) w .centre) where rem₁ : ∀ n → is-n-connected A (2 + n) → ∀ x y → Path (n-Tr A (suc n)) (inc x) (inc y) rem₁ zero a-conn x y = n-Tr-is-hlevel 0 _ _ rem₁ (suc n) a-conn x y = Equiv.from n-Tr-path-equiv (n-Tr-rec (is-hlevel-suc _ (n-Tr-is-hlevel n)) inc (is-n-connected-Tr _ (Path-is-connected (2 + n) a-conn) .centre)) is-connected-+ : ∀ {ℓ} {A : Type ℓ} (n k : Nat) → is-n-connected A (k + n) → is-n-connected A n is-connected-+ n zero w = w is-connected-+ n (suc k) w = is-connected-+ n k (is-connected-suc _ w)

## In terms of propositional truncations🔗

There is an alternative definition of connectedness that avoids talking about arbitrary truncations, and is thus sometimes easier to work with. Generalising the special cases for $n=−1$ (a type is $(−1)-connected$ if and only if it is inhabited) and $n=0$ (a type is $0-connected$ if and only if it is inhabited and all points are merely equal), we can prove that a type is $n-connected$ if and only if it is inhabited and all its path spaces are $(n−1)-connected.$

We can use this to give a definition of connectedness that only makes
use of *propositional* truncations, with the base case being that
all types are
$(n−2)-connected:$

is-n-connected-∥-∥ : ∀ {ℓ} → Type ℓ → Nat → Type ℓ is-n-connected-∥-∥ A zero = Lift _ ⊤ is-n-connected-∥-∥ A (suc n) = ∥ A ∥ × ∀ (a b : A) → is-n-connected-∥-∥ (a ≡ b) n is-n-connected-∥-∥-is-prop : ∀ n → is-prop (is-n-connected-∥-∥ A n) is-n-connected-∥-∥-is-prop zero = hlevel 1 is-n-connected-∥-∥-is-prop (suc n) = ×-is-hlevel 1 (hlevel 1) (Π-is-hlevel² 1 λ _ _ → is-n-connected-∥-∥-is-prop n)

We show that this is equivalent to the $n-truncation$ of a type being contractible, hence in turn to our first definition.

is-contr-n-Tr→∥-∥ : ∀ n → is-contr (n-Tr A (suc n)) → is-n-connected-∥-∥ A (suc n) is-contr-n-Tr→∥-∥ zero h .fst = n-Tr-rec! inc (h .centre) is-contr-n-Tr→∥-∥ zero h .snd = _ is-contr-n-Tr→∥-∥ (suc n) h .fst = n-Tr-rec! inc (h .centre) is-contr-n-Tr→∥-∥ (suc n) h .snd a b = is-contr-n-Tr→∥-∥ n (Equiv→is-hlevel 0 (n-Tr-path-equiv e⁻¹) (Path-is-hlevel 0 h)) ∥-∥→is-contr-n-Tr : ∀ n → is-n-connected-∥-∥ A (suc n) → is-contr (n-Tr A (suc n)) ∥-∥→is-contr-n-Tr zero (a , _) = is-prop∙→is-contr (hlevel 1) (rec! inc a) ∥-∥→is-contr-n-Tr (suc n) (a , h) = case a of λ a → is-prop∙→is-contr (elim! λ a b → Equiv.from n-Tr-path-equiv (∥-∥→is-contr-n-Tr n (h a b) .centre)) (inc a) is-n-connected→∥-∥ : ∀ n → is-n-connected A n → is-n-connected-∥-∥ A n is-n-connected→∥-∥ zero _ = _ is-n-connected→∥-∥ (suc n) h = is-contr-n-Tr→∥-∥ n (n-connected.to n h) ∥-∥→is-n-connected : ∀ n → is-n-connected-∥-∥ A n → is-n-connected A n ∥-∥→is-n-connected zero _ = _ ∥-∥→is-n-connected (suc n) h = n-connected.from n (∥-∥→is-contr-n-Tr n h) is-n-connected≃∥-∥ : ∀ n → is-n-connected A n ≃ is-n-connected-∥-∥ A n is-n-connected≃∥-∥ {A = A} n = prop-ext (is-n-connected-is-prop {A = A} n) (is-n-connected-∥-∥-is-prop n) (is-n-connected→∥-∥ n) (∥-∥→is-n-connected n) module n-connected-∥-∥ {ℓ} {A : Type ℓ} (n : Nat) = Equiv (is-n-connected≃∥-∥ {A = A} n)

## In relation to truncatedness🔗

The following lemmas define
$n-connectedness$
of a type
$A$
by how it can map into the
$n-$*truncated*
types. We then expand this idea to cover the
$n-connected$
maps, too. At the level of types, the “relative” characterisation of
$n-connectedness$
says that, if
$A$
is
$n-connected$
and
$B$
is
$n-truncated,$
then the function
$const:B→(A→B)$
is an equivalence.

The intuitive idea behind this theorem is as follows: we have assumed that $A$ has no interesting information below dimension $n,$ and that $B$ has no interesting information in the dimensions $n$ and above. Therefore, the functions $A→B$ can’t be interesting, since they’re mapping boring data into boring data.

The proof is a direct application of the definition of $n-connectedness$ and the universal property of truncations: $A→B$ is equivalent to $∥A∥_{n}→B,$ but this is equivalent to $B$ since $∥A∥_{n}$ is contractible.

is-n-connected→n-type-const : ∀ n → is-hlevel B (suc n) → is-n-connected A (suc n) → is-equiv {B = A → B} (λ b a → b) is-n-connected→n-type-const {B = B} {A = A} n B-hl A-conn = subst is-equiv (λ i x z → transp (λ i → B) i x) $ snd $ B ≃⟨ Π-contr-eqv (is-n-connected-Tr n A-conn) e⁻¹ ⟩≃ (n-Tr A (suc n) → B) ≃⟨ n-Tr-univ n B-hl ⟩≃ (A → B) ≃∎

This direction of the theorem is actually half of an equivalence. In the other direction, since $∥A∥_{n}$ is $n-truncated$ by definition, we suppose that $const:∥A∥_{n}→(A→∥A∥_{n})$ is an equivalence. From the constructor $inc:A→∥A∥_{n}$ we obtain a point $p:∥A∥_{n},$ and, for all $a,$ we have

$p=const(p)(a)=inc(a).$But by induction on truncation, this says precisely that any $a:∥A∥_{n}$ is equal to $p;$ so $p$ is a centre of contraction, and $A$ is $n-connected.$

n-type-const→is-n-connected : ∀ {ℓ} {A : Type ℓ} n → (∀ {B : Type ℓ} → is-hlevel B (suc n) → is-equiv {B = A → B} (λ b a → b)) → is-n-connected A (suc n) n-type-const→is-n-connected {A = A} n eqv = n-connected.from n $ contr (rem.from inc) $ n-Tr-elim _ (λ h → Path-is-hlevel (suc n) (n-Tr-is-hlevel n)) (rem.ε inc $ₚ_) where module rem = Equiv (_ , eqv {n-Tr A (suc n)} (hlevel _))

We can now generalise this to work with an $n-connected$ map $f:A→B$ and a family $P$ of $n-types$ over $B:$ in this setting, precomposition with $f$ is an equivalence

$(Π_{a:A}P(fa))→(Π_{b:B}P(b)).$This is somewhat analogous to generalising from a recursion principle to an elimination principle. When we were limited to talking about types, we could extend points $b:B$ to functions $A→B,$ but no dependency was possible. With the extra generality, we think of $f$ as including a space of “constructors”, and the equivalence says that exhibiting $P$ at these constructors is equivalent to exhibiting it for the whole type.

relative-n-type-const : (P : B → Type ℓ'') (f : A → B) → ∀ n → is-n-connected-map f n → (∀ x → is-hlevel (P x) n) → is-equiv {A = (∀ b → P b)} {B = (∀ a → P (f a))} (_∘ f)

relative-n-type-const {B = B} {A = A} P f (suc (suc k)) n-conn phl = subst is-equiv (funext λ g → funext λ a → transport-refl _) (rem₁ .snd) where n = suc k open is-iso shuffle : ((b : B) → fibre f b → P b) ≃ ((a : A) → P (f a)) rem₁ : ((b : B) → P b) ≃ ((a : A) → P (f a))

Despite the generality, the proof is just a calculation: observing that we have the following chain of equivalences suffices.

rem₁ = ((b : B) → P b) ≃⟨ Π-cod≃ (λ x → Π-contr-eqv {B = λ _ → P x} (is-n-connected-Tr _ (n-conn x)) e⁻¹) ⟩≃ ((b : B) → n-Tr (fibre f b) (suc n) → P b) ≃⟨ Π-cod≃ (λ x → n-Tr-univ n (phl _)) ⟩≃ ((b : B) → fibre f b → P b) ≃⟨ shuffle ⟩≃ ((a : A) → P (f a)) ≃∎

##
There’s also the `shuffle`

isomorphism that
eliminates the `fibre`

argument using path
induction, but its construction is mechanical.

shuffle = Iso→Equiv λ where .fst g a → g (f a) (a , refl) .snd .inv g b (a , p) → subst P p (g a) .snd .rinv g → funext λ a → transport-refl _ .snd .linv g → funext λ b → funext λ { (a , p) → J (λ b p → subst P p (g (f a) (a , refl)) ≡ g b (a , p)) (transport-refl _) p }

relative-n-type-const {B = B} {A = A} P f 0 n-conn phl = is-contr→is-equiv (Π-is-hlevel 0 phl) (Π-is-hlevel 0 λ _ → phl _) relative-n-type-const {B = B} {A = A} P f 1 n-conn phl = prop-ext (Π-is-hlevel 1 phl) (Π-is-hlevel 1 λ _ → phl _) _ (λ g b → ∥-∥-rec (phl b) (λ (a , p) → subst P p (g a)) (n-conn b)) .snd

We can specialise this to get a literal elimination principle: If $P$ is a family of $n-types$ over an $(1+n)-connected$ pointed type $(A,a_{0}),$ then $P$ holds everywhere if $P(a_{0})$ holds. Moreover, this has the expected computation rule.

module _ n (P : A → Type ℓ) (tr : ∀ x → is-hlevel (P x) n) {a₀ : A} (pa₀ : P a₀) (a-conn : is-n-connected A (suc n)) where connected-elimination-principle : fibre (λ z x → z a₀) (λ _ → pa₀) connected-elimination-principle = relative-n-type-const {A = ⊤} P (λ _ → a₀) n (λ x → retract→is-n-connected n (λ p → tt , p) snd (λ _ → refl) (Path-is-connected n a-conn)) tr .is-eqv (λ _ → pa₀) .centre opaque connected-elim : ∀ x → P x connected-elim = connected-elimination-principle .fst connected-elim-β : connected-elim a₀ ≡ pa₀ connected-elim-β = connected-elimination-principle .snd $ₚ tt

Using these elimination principles, we can prove that a pointed type $(A,a_{0})$ is $(1+n)-connected$ if and only if the inclusion of $a_{0}$ is $n-connected$ when considered as a map $⊤→A.$

is-n-connected-point : ∀ {ℓ} {A : Type ℓ} (a₀ : A) n → is-n-connected-map {A = ⊤} (λ _ → a₀) n → is-n-connected A (suc n) is-n-connected-point {A = A} a₀ n pt-conn = done where rem : ∀ {B : Type ℓ} → is-hlevel B (suc n) → ∀ (f : A → B) a → f a₀ ≡ f a rem b-hl f = equiv→inverse (relative-n-type-const (λ a → f a₀ ≡ f a) _ n pt-conn λ x → Path-is-hlevel' n b-hl _ _) (λ _ → refl) done : is-n-connected A (suc n) done = n-type-const→is-n-connected n λ hl → is-iso→is-equiv $ iso (λ f → f a₀) (λ f → funext λ a → rem hl f a) λ _ → refl point-is-n-connected : ∀ {ℓ} {A : Type ℓ} (a₀ : A) n → is-n-connected A (2 + n) → is-n-connected-map {A = ⊤} (λ _ → a₀) (suc n) point-is-n-connected a₀ n a-conn = connected-elim (suc n) (λ x → is-n-connected (⊤ × (a₀ ≡ x)) (suc n)) (λ x → is-prop→is-hlevel-suc (is-n-connected-is-prop (suc n))) (retract→is-n-connected (suc n) (tt ,_) snd (λ _ → refl) (Path-is-connected {y = a₀} (suc n) a-conn)) a-conn

relative-n-type-const-plus : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} (P : B → Type ℓ'') → (f : A → B) → ∀ n k → is-n-connected-map f n → (∀ x → is-hlevel (P x) (k + n)) → ∀ it → is-hlevel (fibre {A = ((b : B) → P b)} {B = (a : A) → P (f a)} (_∘ f) it) k relative-n-type-const-plus P f n zero f-conn P-hl it = relative-n-type-const P f n f-conn P-hl .is-eqv it relative-n-type-const-plus {A = A} {B = B} P f n (suc k) f-conn P-hl it = done where T = fibre {A = ((b : B) → P b)} {B = (a : A) → P (f a)} (_∘ f) it module _ (gp@(g , p) hq@(h , q) : T) where Q : B → Type _ Q b = g b ≡ h b S = fibre {A = (∀ b → Q b)} {B = (∀ a → Q (f a))} (_∘ f) λ a → happly (p ∙ sym q) a rem₃ : ∀ {ℓ} {A : Type ℓ} {x y : A} (p q : x ≡ y) → (p ≡ q) ≡ (refl ≡ p ∙ sym q) rem₃ {x = x} p q = J (λ y q → (p : x ≡ y) → (p ≡ q) ≡ (refl ≡ p ∙ sym q)) (λ p → sym (ap (refl ≡_) (∙-idr p) ∙ Iso→Path (sym , iso sym (λ _ → refl) (λ _ → refl)))) q p abstract remark : ∀ {h} (α : g ≡ h) {q : h ∘ f ≡ it} → (PathP (λ i → α i ∘ f ≡ it) p q) ≃ (happly α ∘ f ≡ happly (p ∙ sym q)) remark α = J (λ h α → {q : h ∘ f ≡ it} → PathP (λ i → α i ∘ f ≡ it) p q ≃ (happly α ∘ f ≡ happly (p ∙ sym q))) (path→equiv (rem₃ _ _) ∙e (_ , embedding→cancellable {f = happly} λ x → is-contr→is-prop (is-iso→is-equiv (iso funext (λ _ → refl) (λ _ → refl)) .is-eqv x))) α to : Path T gp hq → S to α = happly (ap fst α) , remark (ap fst α) .fst (ap snd α) from : S → Path T gp hq from (a , b) = ap₂ _,_ (funext a) (Equiv.from (remark (funext a)) b) linv : is-left-inverse from to linv x = ap₂ (ap₂ _,_) refl (Equiv.η (remark _) _) done = Path-is-hlevel→is-hlevel k $ λ x y → retract→is-hlevel k (from _ _) (to _ _) (linv _ _) $ relative-n-type-const-plus (Q x y) f n k f-conn (λ x → Path-is-hlevel' (k + n) (P-hl x) _ _) _