4.4 Contractible fibers

4.4 Contractible fibers

Note that our proofs about

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

and

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

made essential use of the fact that the fibers of an

equivalence

are

contractible

.
In fact, it turns out that this property is itself a

sufficient

definition of equivalence.

Definition 4.4.1

(Contractible maps).

A map

f

:

A

→

B

is

contractible

if for all

y

:

B

, the fiber

fib

f

⁢

(

y

)

is contractible.

Thus, the type

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

is defined to be

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

:

≡

∏

y

:

B

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

(

𝖿𝗂𝖻

f

(

y

)

)

(1)

Note that in §3.11 (

http://planetmath.org/311contractibility

) we defined what it means for a

type

to be contractible.
Here we are defining what it means for a

map

to be contractible.
Our terminology follows the general homotopy-theoretic practice of saying that a map has a certain property if all of its (

homotopy

) fibers have that property.
Thus, a type

A

is contractible just when the map

A

→

𝟏

is contractible.
From

http://planetmath.org/node/87580

Chapter 7 onwards we will also call contractible maps and types

(

-

2

)

-truncated

.

We have already shown in

Theorem

4.2.6 (

http://planetmath.org/42halfadjointequivalences#Thmprethm2

) that

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

→

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

.

Conversely

:

Theorem 4.4.2

.

For any

f

:

A

→

B

we have

isContr

⁢

(

f

)

→

ishae

⁢

(

f

)

.

Proof.

Let

P

:

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

. We define an inverse mapping

g

:

B

→

A

by sending each

y

:

B

to the center of

contraction

of the fiber at

y

:

g

(

y

)

:

≡

𝗉𝗋

1

(

𝗉𝗋

1

(

P

y

)

)

We can thus define the homotopy

ϵ

by

mapping

y

to the witness that

g

⁢

(

y

)

indeed belongs to the fiber at

y

:

ϵ

(

y

)

:

≡

𝗉𝗋

2

(

𝗉𝗋

1

(

P

y

)

)

It remains to define

η

and

τ

. This of course amounts to giving an element of

𝗋𝖼𝗈𝗁

f

⁢

(

g

,

ϵ

)

. By Lemma 4.2.11 (

http://planetmath.org/42halfadjointequivalences#Thmprelem5

), this is the same as giving for each

x

:

A

a path from

(

g

⁢

f

⁢

x

,

ϵ

⁢

(

f

⁢

x

)

)

to

(

x

,

𝗋𝖾𝖿𝗅

f

⁢

x

)

in the fiber of

f

over

f

⁢

x

. But this is easy: for any

x

:

A

, the type

𝖿𝗂𝖻

f

⁢

(

f

⁢

x

)

is contractible by

assumption

, hence such a path must exist. We can

construct

it explicitly as

(

𝗉𝗋

2

⁢

(

P

⁢

(

f

⁢

x

)

)

⁢

(

g

⁢

f

⁢

x

,

ϵ

⁢

(

f

⁢

x

)

)

)

-

1

⁢

\centerdot

⁢

(

𝗉𝗋

2

⁢

(

P

⁢

(

f

⁢

x

)

)

⁢

(

x

,

𝗋𝖾𝖿𝗅

f

⁢

x

)

)

.

∎

It is also easy to see:

Lemma 4.4.3

.

For any

f

, the type

isContr

⁢

(

f

)

is a mere proposition.

Proof.

By Lemma 3.11.4 (

http://planetmath.org/311contractibility#Thmprelem2

), each type

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

𝖿𝗂𝖻

f

⁢

(

y

)

)

is a mere proposition.
Thus, by Example 3.6.2 (

http://planetmath.org/36thelogicofmerepropositions#Thmpreeg2

), so is (

1

).
∎

Theorem 4.4.4

.

For any

f

:

A

→

B

we have

isContr

⁢

(

f

)

≃

ishae

⁢

(

f

)

.

Proof.

We have already established a

logical equivalence

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

⇔

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

, and both are mere propositions (Lemma 4.4.3 (

http://planetmath.org/44contractiblefibers#Thmprelem1

),Theorem 4.2.13 (

http://planetmath.org/42halfadjointequivalences#Thmprethm3

)).
Thus, Lemma 3.3.3 (

http://planetmath.org/33merepropositions#Thmprelem2

) applies.
∎

Usually, we prove that a

function

is an equivalence by exhibiting a

quasi-inverse

, but sometimes this definition is more convenient.
For

instance

, it implies that when proving a function to be an equivalence, we are free to assume that its codomain is inhabited.

Corollary 4.4.5

.

If

f

:

A

→

B

is such that

B

→

isequiv

⁢

(

f

)

, then

f

is an equivalence.

Proof.

To show

f

is an equivalence, it suffices to show that

𝖿𝗂𝖻

f

⁢

(

y

)

is contractible for any

y

:

B

.
But if

e

:

B

→

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

, then given any such

y

we have

e

⁢

(

y

)

:

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

, so that

f

is an equivalence and hence

𝖿𝗂𝖻

f

⁢

(

y

)

is contractible, as desired.
∎

Title

4.4 Contractible fibers

\metatable

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