4.9 Univalence implies function extensionality

4.9 Univalence implies function extensionality

In the last

section

of this chapter we include a proof that the univalence axiom implies function
extensionality. Thus, in this section we work

without

the function extensionality axiom.
The proof consists of two steps. First we show
in

Theorem

4.9.4 (

http://planetmath.org/49univalenceimpliesfunctionextensionality#Thmprethm1

) that the univalence
axiom implies a weak form of function extensionality, defined in Definition 4.9.1 (

http://planetmath.org/49univalenceimpliesfunctionextensionality#Thmpredefn1

) below. The
principle of weak function extensionality in turn implies the usual function extensionality,
and it does so without the univalence axiom (Theorem 4.9.5 (

http://planetmath.org/49univalenceimpliesfunctionextensionality#Thmprethm2

)).

Let

𝒰

be a

universe

; we will explicitly indicate where we assume that it is

univalent

.

Definition 4.9.1

.

The

weak function extensionality principle

asserts that there is a

function

(

∏

x

:

A

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

P

⁢

(

x

)

)

)

→

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

∏

x

:

A

P

⁢

(

x

)

)

for any family

P

:

A

→

U

of types over any type

A

.

The following lemma is easy to prove using function extensionality; the point here is that it also follows from univalence without assuming function extensionality separately.

Lemma 4.9.2

.

Assuming

U

is univalent, for any

A

,

B

,

X

:

U

and any

e

:

A

≃

B

, there is an equivalence

(

X

→

A

)

≃

(

X

→

B

)

of which the underlying map is given by post-composition with the underlying function of

e

.

Proof.

As in the proof of Lemma 4.1.1 (

http://planetmath.org/41quasiinverses#Thmprelem1

), we may assume that

e

=

𝗂𝖽𝗍𝗈𝖾𝗊𝗏

⁢

(

p

)

for some

p

:

A

=

B

.
Then by path

induction

, we may assume

p

is

𝗋𝖾𝖿𝗅

A

, so that

e

=

𝗂𝖽

A

.
But in this case, post-composition with

e

is the

identity

, hence an equivalence.
∎

Corollary 4.9.3

.

Let

P

:

A

→

U

be a family of contractible types, i.e.

∏

(

x

:

A

)

isContr

⁢

(

P

⁢

(

x

)

)

.

Then the

projection

pr

1

:

(

∑

(

x

:

A

)

P

⁢

(

x

)

)

→

A

is an equivalence. Assuming

U

is univalent, it follows immediately that precomposition with

pr

1

gives an equivalence

α

:

(

A

→

∑

x

:

A

P

(

x

)

)

≃

(

A

→

A

)

.

Proof.

By Lemma 4.8.1 (

http://planetmath.org/48theobjectclassifier#Thmprelem1

), for

𝗉𝗋

1

:

∑

(

x

:

A

)

P

⁢

(

X

)

→

A

and

x

:

A

we have an equivalence

𝖿𝗂𝖻

𝗉𝗋

1

⁢

(

x

)

≃

P

⁢

(

x

)

.

Therefore

𝗉𝗋

1

is an equivalence whenever each

P

⁢

(

x

)

is

contractible

. The assertion is now a

consequence

of Lemma 4.9.2 (

http://planetmath.org/49univalenceimpliesfunctionextensionality#Thmprelem1

).
∎

In particular, the homotopy fiber of the above equivalence at

𝗂𝖽

A

is contractible. Therefore, we can show that univalence implies weak function extensionality by showing that the dependent function type

∏

(

x

:

A

)

P

⁢

(

x

)

is a retract of

𝖿𝗂𝖻

α

⁢

(

𝗂𝖽

A

)

.

Theorem 4.9.4

.

In a univalent universe

U

, suppose that

P

:

A

→

U

is a family of contractible types
and let

α

be the function of Corollary 4.9.3 (

http://planetmath.org/49univalenceimpliesfunctionextensionality#Thmprecor1

).
Then

∏

(

x

:

A

)

P

⁢

(

x

)

is a retract of

fib

α

⁢

(

id

A

)

. As a consequence,

∏

(

x

:

A

)

P

⁢

(

x

)

is contractible. In other words, the univalence axiom implies the weak function extensionality principle.

Proof.

Define the functions

φ

:

∏

(

x

:

A

)

P

⁢

(

x

)

→

𝖿𝗂𝖻

α

⁢

(

𝗂𝖽

A

)

,

φ

⁢

(

f

)

:

≡

(

λ

x

.

(

x

,

f

(

x

)

)

,

𝗋𝖾𝖿𝗅

𝗂𝖽

A

)

,

and

ψ

:

𝖿𝗂𝖻

α

⁢

(

𝗂𝖽

A

)

→

∏

(

x

:

A

)

P

⁢

(

x

)

,

ψ

⁢

(

g

,

p

)

:

≡

λ

x

.

p

*

(

𝗉𝗋

2

(

g

(

x

)

)

)

.

Then

ψ

⁢

(

φ

⁢

(

f

)

)

=

λ

⁢

x

.

f

⁢

(

x

)

, which is

f

, by the uniqueness principle for dependent function types.
∎

We now show that weak function extensionality implies the usual function extensionality.
Recall from (2.9.2) (

http://planetmath.org/29pitypesandthefunctionextensionalityaxiom#S0.E2

) the function

𝗁𝖺𝗉𝗉𝗅𝗒

(

f

,

g

)

:

(

f

=

g

)

→

(

f

∼

g

)

which
converts equality of functions to homotopy. In the proof that follows, the univalence
axiom is not used.

Theorem 4.9.5

.

Weak function extensionality implies the function extensionality Axiom 2.9.3 (

http://planetmath.org/29pitypesandthefunctionextensionalityaxiom#Thmpreaxiom1

).

Proof.

We want to show that

∏

(

A

:

𝒰

)

∏

(

P

:

A

→

𝒰

)

∏

(

f

,

g

:

∏

(

x

:

A

)

P

(

x

)

)

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

𝗁𝖺𝗉𝗉𝗅𝗒

⁢

(

f

,

g

)

)

.

Since a fiberwise map induces an equivalence on total spaces if and only if it is fiberwise an equivalence by Theorem 4.7.7 (

http://planetmath.org/47closurepropertiesofequivalences#Thmprethm4

), it suffices to show that the function of type

(

∑

g

:

∏

(

x

:

A

)

P

⁢

(

x

)

(

f

=

g

)

)

→

∑

g

:

∏

(

x

:

A

)

P

⁢

(

x

)

(

f

∼

g

)

induced by

λ

(

g

:

∏

(

x

:

A

)

P

(

x

)

)

.

𝗁𝖺𝗉𝗉𝗅𝗒

(

f

,

g

)

is an equivalence.
Since the type on the left is contractible by Lemma 3.11.8 (

http://planetmath.org/311contractibility#Thmprelem5

), it suffices to show that the type on the right:

∑

(

g

:

∏

(

x

:

A

)

P

(

x

)

)

∏

(

x

:

A

)

f

⁢

(

x

)

=

g

⁢

(

x

)

(4.9.6)

is contractible.
Now Theorem 2.15.7 (

http://planetmath.org/215universalproperties#Thmprethm3

) says that this is

equivalent

to

∏

(

x

:

A

)

∑

(

u

:

P

(

x

)

)

f

⁢

(

x

)

=

u

.

(4.9.7)

The proof of Theorem 2.15.7 (

http://planetmath.org/215universalproperties#Thmprethm3

) uses function extensionality, but only for one of the composites.
Thus, without assuming function extensionality, we can conclude that (

4.9.6

) is a retract of (

4.9.7

).
And (

4.9.7

) is a

product

of contractible types, which is contractible by the weak function extensionality principle; hence (

4.9.6

) is also contractible.
∎

Title

4.9 Univalence implies function extensionality

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