HLevel(n) is of hlevel n+1

Authors: Benedikt Ahrens, Chris Kapulkin Title: HLevel(n) is of hlevel n+1 Date: December 2012
In this file we prove the main result of this work: the type of types of hlevel n is itself of hlevel n+1.

Weak equivalence between identity types in HLevel and U

To show that HLevel(n) is of hlevel n + 1, we need to show that its path spaces are of hlevel n.
First, we show that each of its path spaces equivalent to the path space of the underlying types.
More generally, we prove this for any predicate P : UU hProp which we later instantiate with P := isofhlevel n.
Overview of the proof: Identity of Sigmas <~> Sigma of Identities <~> Identities in U , where the first equivalence is called weq1 and the second one is called weq2.

Local Lemma weq1 (P : UU hProp) (X X' : UU)
(p : P X) (p' : P X')
: (X,, p) = (X',, p' : (T : UU), P T )

( w : X = X', transportf P w p = p').
Proof.
apply total2_paths_equiv.
Defined.

This helper lemma is needed to show that our fibration is indeed a predicate, so that we can instantiate the hProposition P with this fibration.

Local Lemma ident_is_prop (P : UU hProp) (X X' : UU)
(p : P X) (p' : P X') (w : X = X')
: isaprop (transportf P w p = p').
Proof.
apply isapropifcontr.
apply (pr2 (P X')).
Defined.

We construct the equivalence weq2 as a projection from a total space, which, by the previous lemma, is a weak equivalence.

Local Lemma weq2 (P : UU hProp) (X X' : UU)
(p : P X) (p' : P X')
: ( w : X = X', transportf P w p = p')

(X = X').
Proof.
use weqpr1.
intro. cbn. apply (pr2 (P X')).
Defined.

Composing weq1 and weq2 yields the desired weak equivalence.

Local Lemma Id_p_weq_Id (P : UU hProp) (X X' : UU)
(p : P X) (p' : P X')
: (X ,, p) = (X',, p' : T , P T)

X = X'.
Proof.
set (f := weq1 P X X' p p').
set (g := weq2 P X X' p p').
set (fg := weqcomp f g).
exact fg.
Defined.

Hlevel of path spaces

We show that if X and X' are of hlevel n, then so is their identity type X = X'. The proof works differently for n = 0 and n = S n', so we treat these two cases in separate lemmas isofhlevel0pathspace and isofhlevelSnpathspace and put them together in the lemma isofhlevelpathspace.

The case n=0

Lemma iscontr_weq (X Y : UU)
: iscontr X iscontr Y iscontr (X Y).
Proof.
intros cX cY.
(weqcontrcontr cX cY ).
intro f.
apply subtypePath.
{ exact isapropisweq. }
apply funextfun. cbn. intro x. apply (pr2 cY).
Defined.

Lemma isofhlevel0pathspace (X Y : UU)
: iscontr X iscontr Y iscontr (X = Y).
Proof.
intros pX pY.
set (H := isofhlevelweqb 0 (eqweqmap ,, univalenceAxiom X Y)).
apply H. clear H.
apply iscontr_weq;
assumption.
Defined.

The case n=Sn'

Lemma isofhlevelSnpathspace : n : nat, X Y : UU,
isofhlevel (S n) Y isofhlevel (S n) (X = Y).
Proof.
intros n X Y pY.
set (H:=isofhlevelweqb (S n) (eqweqmap ,, univalenceAxiom X Y)).
apply H.
apply isofhlevelsnweqtohlevelsn.
assumption.
Defined.

The lemma itself

Lemma isofhlevelpathspace : n : nat, X Y : UU,
isofhlevel n X isofhlevel n Y isofhlevel n (X = Y).
Proof.
intros n.
induction n as [| n _ ].
- intros X Y pX pY.
apply isofhlevel0pathspace;
assumption.
- intros.
apply isofhlevelSnpathspace;
assumption.
Defined.

HLevel

We define the type HLevel n of types of hlevel n.

Definition HLevel (n : nat) : UU := X : UU, isofhlevel n X.

Main theorem: HLeveln is of hlevel Sn

Lemma isofhlevel_HLevel (n : nat) : isofhlevel (S n) (HLevel n).
Proof.
cbn.
intros X X'.
induction X as [X p].
induction X' as [X' p'].
set (H := isofhlevelweqb n
(Id_p_weq_Id (λ X, (isofhlevel n X,, isapropisofhlevel _ _)) X X' p p')).
cbn in H.
apply H.
apply isofhlevelpathspace;
assumption.
Defined.

Aside: Univalence for predicates and hlevels

As a corollary from Id_p_weq_Id, we obtain a version of the Univalence Axiom for predicates on the universe U. In particular, we can instantiate this predicate with isofhlevel n.

Lemma UA_for_Predicates (P : UU hProp) (X X' : UU)
(pX : P X) (pX' : P X') :
(tpair _ X pX) = (tpair P X' pX') (X X').
Proof.
set (f := Id_p_weq_Id P X X' pX pX').
set (g := tpair _ _ (univalenceAxiom X X')).
exact (weqcomp f g).
Defined.

Corollary UA_for_HLevels : (n : nat) (X X' : HLevel n),
(X = X') (pr1 X pr1 X').
Proof.
intros n [X pX] [X' pX'].
simpl.
apply (UA_for_Predicates
(λ X, tpair isaprop (isofhlevel n X)
(isapropisofhlevel _ _))).
Defined.