4.7 Closure properties of equivalences

4.7 Closure properties of equivalences

We have already seen in Lemma 2.4.12 (

http://planetmath.org/http://planetmath.org/24homotopiesandequivalences#Thmprelem3

) that

equivalences

are

closed under

composition

.
Furthermore, we have:

Theorem 4.7.1

(The 2-out-of-3 property).

Suppose

f

:

A

→

B

and

g

:

B

→

C

.
If any two of

f

,

g

, and

g

∘

f

are equivalences, so is the third.

Proof.

If

g

∘

f

and

g

are equivalences, then

(

g

∘

f

)

-

1

∘

g

is a

quasi-inverse

to

f

.
On the one hand, we have

(

g

∘

f

)

-

1

∘

g

∘

f

∼

𝗂𝖽

A

, while on the other we have

f

∘

(

g

∘

f

)

-

1

∘

g

∼

g

-

1

∘

g

∘

f

∘

(

g

∘

f

)

-

1

∘

g

∼

g

-

1

∘

g

∼

𝗂𝖽

B

.

Similarly, if

g

∘

f

and

f

are equivalences, then

f

∘

(

g

∘

f

)

-

1

is a quasi-inverse to

g

.
∎

This is a standard closure condition on equivalences from

homotopy theory

.
Also well-known is that they are closed under

retracts

, in the following sense.

Definition 4.7.2

.

A function

g

:

A

→

B

is said to be a

retract

of a function

f

:

X

→

Y

if there is a

diagram

\includegraphics

HoTT_fig_4.7.1.png

for which there are

1.

a

homotopy

R

:

r

∘

s

∼

𝗂𝖽

A

.

2.

a homotopy

R

′

:

r

′

∘

s

′

∼

𝗂𝖽

B

.

3.

a homotopy

L

:

f

∘

s

∼

s

′

∘

g

.

4.

a homotopy

K

:

g

∘

r

∼

r

′

∘

f

.

5.

for every

a

:

A

, a path

H

⁢

(

a

)

witnessing the

commutativity

of the square

\includegraphics

HoTT_fig_4.7.2.png

Recall that in §3.11 (

http://planetmath.org/311contractibility

) we defined what it means for a type to be a retract of another.
This is a special case of the above definition where

B

and

Y

are

𝟏

.

Conversely

, just as with

contractibility

,

retractions

of maps induce retractions of their fibers.

Lemma 4.7.3

.

If a function

g

:

A

→

B

is a retract of a function

f

:

X

→

Y

, then

fib

g

⁢

(

b

)

is a retract of

fib

f

⁢

(

s

′

⁢

(

b

)

)

for every

b

:

B

, where

s

′

:

B

→

Y

is as in Definition 4.7.2 (

http://planetmath.org/47closurepropertiesofequivalences#Thmpredefn1

).

Proof.

Suppose that

g

:

A

→

B

is a retract of

f

:

X

→

Y

. Then for any

b

:

B

we have the functions

φ

b

:

𝖿𝗂𝖻

g

⁢

(

b

)

→

𝖿𝗂𝖻

f

⁢

(

s

′

⁢

(

b

)

)

,

φ

b

⁢

(

a

,

p

)

:

≡

(

s

(

a

)

,

L

(

a

)

\centerdot

s

′

(

p

)

)

,

ψ

b

:

𝖿𝗂𝖻

f

⁢

(

s

′

⁢

(

b

)

)

→

𝖿𝗂𝖻

g

⁢

(

b

)

,

ψ

b

⁢

(

x

,

q

)

:

≡

(

r

(

x

)

,

K

(

x

)

\centerdot

r

′

(

q

)

\centerdot

R

′

(

b

)

)

.

Then we have

ψ

b

(

φ

b

(

a

,

p

)

)

≡

(

r

(

s

(

a

)

)

,

K

(

s

(

a

)

)

\centerdot

r

′

(

L

(

a

)

\centerdot

s

′

(

p

)

)

\centerdot

R

′

(

b

)

)

.
We claim

ψ

b

is a retraction with

section

φ

b

for all

b

:

B

, which is to say that for all

(

a

,

p

)

:

𝖿𝗂𝖻

g

⁢

(

b

)

we have

ψ

b

(

φ

b

(

a

,

p

)

)

=

(

a

,

p

)

.
In other words, we want to show

∏

(

b

:

B

)

∏

(

a

:

A

)

∏

(

p

:

g

(

a

)

=

b

)

ψ

b

(

φ

b

(

a

,

p

)

)

=

(

a

,

p

)

.

By reordering the first two

Π

s and applying a version of Lemma 3.11.9 (

http://planetmath.org/311contractibility#Thmprelem6

), this is

equivalent

to

∏

a

:

A

ψ

g

⁢

(

a

)

(

φ

g

⁢

(

a

)

(

a

,

𝗋𝖾𝖿𝗅

g

⁢

(

a

)

)

)

=

(

a

,

𝗋𝖾𝖿𝗅

g

⁢

(

a

)

)

.

For any

a

, by

Theorem

2.7.2 (

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

), this equality of pairs is equivalent to a pair of equalities. The first

components

are equal by

R

⁢

(

a

)

:

r

⁢

(

s

⁢

(

a

)

)

=

a

, so we need only show

R

(

a

)

*

(

K

(

s

(

a

)

)

\centerdot

r

′

(

L

(

a

)

)

\centerdot

R

′

(

g

(

a

)

)

)

=

𝗋𝖾𝖿𝗅

g

⁢

(

a

)

.

But this transportation computes as

g

⁢

(

R

⁢

(

a

)

)

-

1

⁢

\centerdot

⁢

K

⁢

(

s

⁢

(

a

)

)

⁢

\centerdot

⁢

r

′

⁢

(

L

⁢

(

a

)

)

⁢

\centerdot

⁢

R

′

⁢

(

g

⁢

(

a

)

)

, so the required path is given by

H

⁢

(

a

)

.
∎

Theorem 4.7.4

.

If

g

is a retract of an equivalence

f

, then

g

is also an equivalence.

Proof.

By Lemma 4.7.3 (

http://planetmath.org/47closurepropertiesofequivalences#Thmprelem1

), every fiber of

g

is a retract of a fiber of

f

.
Thus, by Lemma 3.11.7 (

http://planetmath.org/311contractibility#Thmprelem4

), if the latter are all

contractible

, so are the former.
∎

Finally, we show that fiberwise equivalences can be characterized in terms of equivalences of total spaces.
To explain the terminology, recall from §2.3 (

http://planetmath.org/23typefamiliesarefibrations

) that a type family

P

:

A

→

𝒰

can be viewed as a

fibration

over

A

with total space

∑

(

x

:

A

)

P

⁢

(

x

)

, the fibration being is the

projection

𝗉𝗋

1

:

∑

(

x

:

A

)

P

⁢

(

x

)

→

A

.
From this point of view, given two type families

P

,

Q

:

A

→

𝒰

, we may refer to a function

f

:

∏

(

x

:

A

)

(

P

(

x

)

→

Q

(

x

)

)

as a

fiberwise map

or a

fiberwise

transformation

.

Such a map induces a function on total spaces:

Definition 4.7.5

.

Given type families

P

,

Q

:

A

→

U

and a map

f

:

∏

(

x

:

A

)

P

⁢

(

x

)

→

Q

⁢

(

x

)

, we define

𝗍𝗈𝗍𝖺𝗅

(

f

)

:

≡

λ

w

.

(

𝗉𝗋

1

w

,

f

(

𝗉𝗋

1

w

,

𝗉𝗋

2

w

)

)

:

∑

x

:

A

P

(

x

)

→

∑

x

:

A

Q

(

x

)

.

Theorem 4.7.6

.

Suppose that

f

is a fiberwise transformation between families

P

and

Q

over a type

A

and let

x

:

A

and

v

:

Q

⁢

(

x

)

. Then we have an equivalence

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

(

(

x

,

v

)

)

≃

𝖿𝗂𝖻

f

⁢

(

x

)

(

v

)

.

Proof.

We calculate:

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

(

(

x

,

v

)

)

≡

∑

w

:

∑

(

x

:

A

)

P

⁢

(

x

)

(

𝗉𝗋

1

w

,

f

(

𝗉𝗋

1

w

,

𝗉𝗋

2

w

)

)

=

(

x

,

v

)

≃

∑

(

a

:

A

)

∑

(

u

:

P

(

a

)

)

(

a

,

f

(

a

,

u

)

)

=

(

x

,

v

)

(by http://planetmath.org/node/87641Exercise 2.10)

≃

∑

(

a

:

A

)

∑

(

u

:

P

(

a

)

)

∑

(

p

:

a

=

x

)

p

*

(

f

(

a

,

u

)

)

=

v

(by Theorem 0.1 (http://planetmath.org/27sigmatypes0.Thmthm1))

≃

∑

(

a

:

A

)

∑

(

p

:

a

=

x

)

∑

(

u

:

P

(

a

)

)

p

*

(

f

(

a

,

u

)

)

=

v

≃

∑

u

:

P

⁢

(

x

)

f

⁢

(

x

,

u

)

=

v

($*$)

≡

𝖿𝗂𝖻

f

⁢

(

x

)

⁢

(

v

)

.

The equivalence (

($*$)

‣

4.7 Closure properties of equivalences

) follows from Lemma 3.11.9 (

http://planetmath.org/311contractibility#Thmprelem6

),Lemma 3.11.8 (

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

),

http://planetmath.org/node/87641

Exercise 2.10.
∎

We say that a fiberwise transformation

f

:

∏

(

x

:

A

)

P

⁢

(

x

)

→

Q

⁢

(

x

)

is a

fiberwise equivalence

if each

f

⁢

(

x

)

:

P

⁢

(

x

)

→

Q

⁢

(

x

)

is an equivalence.

Theorem 4.7.7

.

Suppose that

f

is a fiberwise transformation between families

P

and

Q

over a type

A

.
Then

f

is a fiberwise equivalence if and only if

total

⁢

(

f

)

is an equivalence.

Proof.

Let

f

,

P

,

Q

and

A

be as in the statement of the theorem.
By Theorem 4.7.6 (

http://planetmath.org/47closurepropertiesofequivalences#Thmprethm3

) it follows for all

x

:

A

and

v

:

Q

⁢

(

x

)

that

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

(

(

x

,

v

)

)

is contractible if and only if

𝖿𝗂𝖻

f

⁢

(

x

)

⁢

(

v

)

is contractible.
Thus,

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

⁢

(

w

)

is contractible for all

w

:

∑

(

x

:

A

)

Q

⁢

(

x

)

if and only if

𝖿𝗂𝖻

f

⁢

(

x

)

⁢

(

v

)

is contractible for all

x

:

A

and

v

:

Q

⁢

(

x

)

.
∎

Title

4.7 Closure properties of equivalences

\metatable

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