open import 1Lab.HLevel.Retracts
open import 1Lab.Type.Sigma
open import 1Lab.Univalence
open import 1Lab.HLevel
open import 1Lab.Equiv
open import 1Lab.Path
open import 1Lab.Type

module 1Lab.HLevel.Universe where

Universes of n-types🔗

A common phenomenon in higher category theory is that the collection of all nn-categories in a given universe \ell assembles into an (n+1)(n+1)-category in the successor universe 1+1+\ell:

  • The collection of all sets (0-categories) is a (1-)-category;
  • The collection of all (1-)categories is a 2-category;
  • The collection of all 2-categories is a 3-category;

Because of the univalence axiom, the same phenomenon can be observed in homotopy type theory: The subuniverse of \ell consisting of all nn-types is a (n+1)(n+1)-type in 1+1+\ell. That means: the universe of propositions is a set, the universe of sets is a groupoid, the universe of groupoids is a bigroupoid, and so on.

h-Levels of Equivalences🔗

As warmup, we prove that if AA and BB are nn-types, then so is the type of equivalences ABA \simeq B. For the case where nn is a successor, this only depends on the h-level of BB.

≃-is-hlevel : (n : Nat)  is-hlevel A n  is-hlevel B n  is-hlevel (A  B) n
≃-is-hlevel {A = A} {B = B} zero Ahl Bhl = contr (f , f-eqv) deform where
  f : A  B
  f _ = Bhl .centre

  f-eqv : is-equiv f
  f-eqv = is-contr→is-equiv Ahl Bhl

For the zero case, we’re asked to give a proof of contractibility of A ≃ B. As the centre we pick the canonical function sending xx to the centre of contraction of BB, which is an equivalence because it is a map between contractible types.

By the characterisation of paths in Σ and the fact that being an equivalence is a proposition, we get the required family of paths deforming any ABA \simeq B to our f.

  deform : (g : A  B)  (f , f-eqv)  g
  deform (g , g-eqv) = Σ-path  i x  Bhl .paths (g x) i)
                              (is-equiv-is-prop _ _ _)

As mentioned before, the case for successors does not depend on the proof that AA has the given h-level. This is because, for n1n \ge 1, ABA \simeq B has the same h-level as ABA \to B, which is the same as BB.

≃-is-hlevel (suc n) _ Bhl =
  Σ-is-hlevel (suc n)
    (fun-is-hlevel (suc n) Bhl)
    λ f  is-prop→is-hlevel-suc (is-equiv-is-prop f)

h-Levels of Paths🔗

Univalence states that the type XYX ≡ Y is equivalent to X YX \simeq Y. Since the latter is of h-level nn when XX and YY are nn-types, then so is the former:

≡-is-hlevel : (n : Nat)  is-hlevel A n  is-hlevel B n  is-hlevel (A  B) n
≡-is-hlevel n Ahl Bhl = equiv→is-hlevel n ua univalence⁻¹ (≃-is-hlevel n Ahl Bhl)


We refer to the dependent sum of the family is-hlevel - n as n-Type:

record n-Type  n : Type (lsuc ) where
  constructor _,_
    ∣_∣   : Type 
    is-tr : is-hlevel ∣_∣ n
  infix 100 ∣_∣
    H-Level-n-type :  {k}  H-Level ∣_∣ (n + k)
    H-Level-n-type = basic-instance n is-tr

open n-Type using (∣_∣ ; is-tr) public

Like mentioned in the introduction, the main theorem of this section is that n-Type is a type of h-level n+1n+1. We take a detour first and establish a characterisation of paths for n-Type: Since is-tr is a proposition, paths in n-Type are determined by paths of the underlying types. By univalence, these correspond to equivalences of the underlying type:

n-ua : {n : Nat} {X Y : n-Type  n}   X    Y   X  Y
n-ua f i .∣_∣ = ua f i
n-ua {n = n} {X} {Y} f i .is-tr =
  is-prop→pathp  i  is-hlevel-is-prop {A = ua f i} n)
    (X .is-tr) (Y .is-tr) i

n-univalence : {n : Nat} {X Y : n-Type  n}  ( X    Y )  (X  Y)
n-univalence {n = n} {X} {Y} = n-ua , is-iso→is-equiv isic where
  inv :  {Y}  X  Y   X    Y 
  inv p = path→equiv (ap ∣_∣ p)

  linv :  {Y}  is-left-inverse (inv {Y}) n-ua
  linv x = Σ-prop-path is-equiv-is-prop (funext λ x  transport-refl _)

  rinv :  {Y}  is-right-inverse (inv {Y}) n-ua
  rinv = J  y p  n-ua (inv p)  p) path where
    path : n-ua (inv {X} refl)  refl
    path i j .∣_∣ = equiv→retraction (univalence {A =  X }) refl i j
    path i j .is-tr = is-prop→squarep
       i j  is-hlevel-is-prop
        {A = equiv→retraction (univalence {A =  X }) refl i j} n)
       j  X .is-tr)  j  n-ua {X = X} {Y = X} (path→equiv refl) j .is-tr)
       j  X .is-tr)  j  X .is-tr)
      i j

  isic : is-iso n-ua
  isic = iso inv rinv (linv {Y})

Since h-levels are closed under equivalence, and we already have an upper bound on the h-level of XYX \simeq Y when YY is an nn-type, we know that nn-Type is a (n+1)(n+1)-type:

n-Type-is-hlevel :  n  is-hlevel (n-Type  n) (suc n)
n-Type-is-hlevel zero x y = n-ua
  ((λ _  y .is-tr .centre) , is-contr→is-equiv (x .is-tr) (y .is-tr))
n-Type-is-hlevel (suc n) x y =
  is-hlevel≃ (suc n) n-univalence (≃-is-hlevel (suc n) (x .is-tr) (y .is-tr))

For 1-categorical mathematics, the important h-levels are the propositions and the sets, so they get short names:

Set :    Type (lsuc )
Set  = n-Type  2

Prop :    Type (lsuc )
Prop  = n-Type  1