2.10 Universes and the univalence axiom

2.10 Universes and the univalence axiom

Given two types

A

and

B

, we may consider them as elements of some

universe

type

𝒰

, and thereby form the identity type

A

=

𝒰

B

.
As mentioned in the

introduction

,

univalence

is the identification of

A

=

𝒰

B

with the type

(

A

≃

B

)

of

equivalences

from

A

to

B

, which we described in §2.4 (

http://planetmath.org/24homotopiesandequivalences

).
We perform this identification by way of the following canonical

function

.

Lemma 2.10.1

.

For types

A

,

B

:

U

, there is a certain function,

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

:

(

A

=

𝒰

B

)

→

(

A

≃

B

)

,

(2.10.1)

defined in the proof.

Proof.

We could construct this directly by

induction

on equality, but the following description is more convenient.
Note that the

identity function

𝗂𝖽

𝒰

:

𝒰

→

𝒰

may be regarded as a type family

indexed by

the universe

𝒰

; it assigns to each type

X

:

𝒰

the type

X

itself.
(When regarded as a

fibration

, its total space is the type

∑

(

A

:

𝒰

)

A

of “pointed types”; see also §4.8 (

http://planetmath.org/48theobjectclassifier

).)
Thus, given a path

p

:

A

=

𝒰

B

, we have a transport function

p

*

:

A

→

B

.
We claim that

p

*

is an equivalence.
But by induction, it suffices to assume that

p

is

𝗋𝖾𝖿𝗅

A

, in which case

p

*

≡

𝗂𝖽

A

, which is an equivalence by Example 2.4.7 (

http://planetmath.org/24homotopiesandequivalences#Thmpreeg1

).
Thus, we can define

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

p

)

to be

p

*

(together with the above proof that it is an equivalence).
∎

We would like to say that

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

is an equivalence.
However, as with

𝗁𝖺𝗉𝗉𝗅𝗒

for function types, the

type theory

described in

http://planetmath.org/node/87533

Chapter 1 is insufficient to guarantee this.
Thus, as we did for function extensionality, we formulate this property as an axiom: Voevodsky’s

univalence axiom

.

Axiom 2.10.3

(Univalence).

For any

A

,

B

:

U

, the function (

2.10.1

) is an equivalence,

(

A

=

𝒰

B

)

≃

(

A

≃

B

)

.

Technically, the univalence axiom is a statement about a particular universe type

𝒰

.
If a universe

𝒰

satisfies

this axiom, we say that it is

univalent

.
Except when otherwise noted (e.g. in §4.9 (

http://planetmath.org/49univalenceimpliesfunctionextensionality

)) we will assume that

all

universes are univalent.

Remark 2.10.4

.

It is important for the univalence axiom that we defined

A

≃

B

using a “good” version of

isequiv

as described in §2.4 (

http://planetmath.org/24homotopiesandequivalences

), rather than (say) as

∑

(

f

:

A

→

B

)

qinv

⁢

(

f

)

.

In particular, univalence means that

equivalent

types may be identified

.
As we did in previous

sections

, it is useful to break this equivalence into:

•

An introduction rule for (

A

=

𝒰

B

),

𝗎𝖺

:

(

A

≃

B

)

→

(

A

=

𝒰

B

)

.

•

The elimination rule, which is

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

,

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

≡

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

X

↦

X

:

(

A

=

𝒰

B

)

→

(

A

≃

B

)

.

•

The propositional

computation

rule,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

X

↦

X

⁢

(

𝗎𝖺

⁢

(

f

)

,

x

)

=

f

⁢

(

x

)

.

•

The propositional uniqueness principle:
for any

p

:

A

=

B

,

p

=

𝗎𝖺

⁢

(

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

X

↦

X

⁢

(

p

)

)

.

We can also identify the

reflexivity

,

concatenation

, and

inverses

of equalities in the universe with the corresponding

operations

on equivalences:

𝗋𝖾𝖿𝗅

A

=

𝗎𝖺

⁢

(

𝗂𝖽

A

)

𝗎𝖺

⁢

(

f

)

⁢

\centerdot

⁢

𝗎𝖺

⁢

(

g

)

=

𝗎𝖺

⁢

(

g

∘

f

)

𝗎𝖺

⁢

(

f

)

-

1

=

𝗎𝖺

⁢

(

f

-

1

)

.

The first of these follows because

𝗂𝖽

A

=

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

𝗋𝖾𝖿𝗅

A

)

by definition of

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

, and

𝗎𝖺

is the inverse of

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

.
For the second, if we define

p

:

≡

𝗎𝖺

(

f

)

and

q

:

≡

𝗎𝖺

(

g

)

, then we have

𝗎𝖺

⁢

(

g

∘

f

)

=

𝗎𝖺

⁢

(

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

q

)

∘

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

p

)

)

=

𝗎𝖺

⁢

(

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

p

⋅

q

)

)

=

p

⋅

q

using Lemma 2.3.9 (

http://planetmath.org/23typefamiliesarefibrations#Thmprelem6

) and the definition of

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

.
The third is similar.

The following observation, which is a special case of Lemma 2.3.10 (

http://planetmath.org/23typefamiliesarefibrations#Thmprelem7

), is often useful when applying the univalence axiom.

Lemma 2.10.5

.

For any type family

B

:

A

→

U

and

x

,

y

:

A

with a path

p

:

x

=

y

and

u

:

B

⁢

(

x

)

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

p

,

u

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

X

↦

X

⁢

(

𝖺𝗉

B

⁢

(

p

)

,

u

)

=

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

𝖺𝗉

B

⁢

(

p

)

)

⁢

(

u

)

.

Title

2.10 Universes and the univalence axiom

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