4.8 The object classifier

4.8 The object classifier

In

type theory

we have a basic notion of

family of types

, namely a

function

B

:

A

→

𝒰

.
We have seen that such families behave somewhat like

fibrations

in

homotopy theory

, with the fibration being the

projection

𝗉𝗋

1

:

∑

(

a

:

A

)

B

⁢

(

a

)

→

A

.
A basic fact in homotopy theory is that every map is

equivalent

to a fibration.
With univalence at our disposal, we can prove the same thing in type theory.

Lemma 4.8.1

.

For any type family

B

:

A

→

U

, the fiber of

pr

1

:

∑

(

x

:

A

)

B

⁢

(

x

)

→

A

over

a

:

A

is equivalent to

B

⁢

(

a

)

:

𝖿𝗂𝖻

𝗉𝗋

1

⁢

(

a

)

≃

B

⁢

(

a

)

Proof.

We have

𝖿𝗂𝖻

𝗉𝗋

1

⁢

(

a

)

:

≡

∑

u

:

∑

(

x

:

A

)

B

⁢

(

x

)

𝗉𝗋

1

(

u

)

=

a

≃

∑

(

x

:

A

)

∑

(

b

:

B

(

x

)

)

(

x

=

a

)

≃

∑

(

x

:

A

)

∑

(

p

:

x

=

a

)

B

⁢

(

x

)

≃

B

⁢

(

a

)

using the left

universal property

of identity types.
∎

Lemma 4.8.2

.

For any function

f

:

A

→

B

, we have

A

≃

∑

(

b

:

B

)

fib

f

⁢

(

b

)

.

Proof.

We have

∑

b

:

B

𝖿𝗂𝖻

f

⁢

(

b

)

:

≡

∑

(

b

:

B

)

∑

(

a

:

A

)

(

f

(

a

)

=

b

)

≃

∑

(

a

:

A

)

∑

(

b

:

B

)

(

f

(

a

)

=

b

)

≃

A

using the fact that

∑

(

b

:

B

)

(

f

(

a

)

=

b

)

is

contractible

.
∎

Theorem 4.8.3

.

For any type

B

there is an

equivalence

χ

:

(

∑

A

:

𝒰

(

A

→

B

)

)

≃

(

B

→

𝒰

)

.

Proof.

We have to

construct

quasi-inverses

χ

:

(

∑

A

:

𝒰

(

A

→

B

)

)

→

B

→

𝒰

ψ

:

(

B

→

𝒰

)

→

(

∑

A

:

𝒰

(

A

→

B

)

)

.

We define

χ

by

χ

(

(

A

,

f

)

,

b

)

:

≡

𝖿𝗂𝖻

f

(

b

)

, and

ψ

by

ψ

(

P

)

:

≡

(

(

∑

(

b

:

B

)

P

(

b

)

)

,

𝗉𝗋

1

)

.
Now we have to verify that

χ

∘

ψ

∼

𝗂𝖽

and that

ψ

∘

χ

∼

𝗂𝖽

.

1.

Let

P

:

B

→

𝒰

.
By Lemma 4.8.1 (

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

),

𝖿𝗂𝖻

𝗉𝗋

1

⁢

(

b

)

≃

P

⁢

(

b

)

for any

b

:

B

, so it follows immediately
that

P

∼

χ

⁢

(

ψ

⁢

(

P

)

)

.

2.

Let

f

:

A

→

B

be a function. We have to find a path

(

∑

(

b

:

B

)

𝖿𝗂𝖻

f

(

b

)

,

𝗉𝗋

1

)

=

(

A

,

f

)

.

First note that by Lemma 4.8.2 (

http://planetmath.org/48theobjectclassifier#Thmprelem2

), we have

e

:

∑

(

b

:

B

)

𝖿𝗂𝖻

f

⁢

(

b

)

≃

A

with

e

(

b

,

a

,

p

)

:

≡

a

and

e

-

1

(

a

)

:

≡

(

f

(

a

)

,

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

.
By

Theorem

2.7.2 (

http://planetmath.org/27sigmatypes#Thmprethm1

), it remains to show

(

𝗎𝖺

(

e

)

)

*

(

𝗉𝗋

1

)

=

f

.
But by the computation rule for univalence and (2.9.4) (

http://planetmath.org/29pitypesandthefunctionextensionalityaxiom#S0.E3

), we have

(

𝗎𝖺

(

e

)

)

*

(

𝗉𝗋

1

)

=

𝗉𝗋

1

∘

e

-

1

, and the definition of

e

-

1

immediately yields

𝗉𝗋

1

∘

e

-

1

≡

f

.∎

In particular, this implies that we have an

object classifier

in the sense of higher topos theory.
Recall from Definition 2.1.7 (

http://planetmath.org/21typesarehighergroupoids#Thmpredefn1

) that

𝒰

∙

denotes the type

∑

(

A

:

𝒰

)

A

of pointed types.

Theorem 4.8.4

.

Let

f

:

A

→

B

be a function. Then the

diagram

\includegraphics

HoTT_fig_4.8.1.png

is a

pullback square

(see

http://planetmath.org/node/87642

Exercise 2.11).
Here the function

ϑ

f

is defined by

λ

a

.

(

𝖿𝗂𝖻

f

(

f

(

a

)

)

,

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

)

.

Proof.

Note that we have the equivalences

A

≃

∑

b

:

B

𝖿𝗂𝖻

f

⁢

(

b

)

≃

∑

(

b

:

B

)

∑

(

X

:

𝒰

)

∑

(

p

:

𝖿𝗂𝖻

f

(

b

)

=

X

)

X

≃

∑

(

b

:

B

)

∑

(

X

:

𝒰

)

∑

(

x

:

X

)

𝖿𝗂𝖻

f

⁢

(

b

)

=

X

≃

∑

(

b

:

B

)

∑

(

Y

:

𝒰

∙

)

𝖿𝗂𝖻

f

⁢

(

b

)

=

𝗉𝗋

1

⁢

Y

≡

B

×

𝒰

𝒰

∙

.

which gives us a composite equivalence

e

:

A

≃

B

×

𝒰

𝒰

∙

.
We may display the action of this composite equivalence step by step by

a

↦

(

f

(

a

)

,

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

)

↦

(

f

(

a

)

,

𝖿𝗂𝖻

f

(

f

(

a

)

)

,

𝗋𝖾𝖿𝗅

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

,

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

)

↦

(

f

(

a

)

,

𝖿𝗂𝖻

f

(

f

(

a

)

)

,

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

,

𝗋𝖾𝖿𝗅

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

)

.

Therefore, we get

homotopies

f

∼

𝗉𝗋

1

∘

e

and

ϑ

f

∼

𝗉𝗋

2

∘

e

.
∎

Title

4.8 The object classifier

\metatable

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