7.5 Connectedness

7.5 Connectedness

An

n

-type is one that has no interesting information above dimension

n

.
By contrast, an

n

-connected type

is one that has no interesting information

below

dimension

n

.
It turns out to be natural to study a more general notion for

functions

as well.

Definition 7.5.1

.

A function

f

:

A

→

B

is said to be

n

-connected

if for all

b

:

B

, the type

∥

fib

f

⁢

(

b

)

∥

n

is

contractible

:

𝖼𝗈𝗇𝗇

n

(

f

)

:

≡

∏

b

:

B

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

(

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

)

.

A type

A

is said to be

n

-connected

if the unique function

A

→

1

is

n

-connected, i.e. if

∥

A

∥

n

is contractible.

Thus, a function

f

:

A

→

B

is

n

-connected if and only if

𝖿𝗂𝖻

f

⁢

(

b

)

is

n

-connected for every

b

:

B

.
Of course, every function is

(

-

2

)

-connected.
At the next level, we have:

Lemma 7.5.2

.

A function

f

is

(

-

1

)

-connected if and only if it is

surjective

in the sense of §4.6 (

http://planetmath.org/46surjectionsandembeddings

).

Proof.

We defined

f

to be surjective if

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

-

1

is inhabited for all

b

.
But since it is a mere proposition, inhabitation is

equivalent

to

contractibility

.
∎

Thus,

n

-connectedness of a function for

n

≥

0

can be thought of as a strong form of surjectivity.
Category-theoretically,

(

-

1

)

-connectedness corresponds to essential surjectivity on objects, while

n

-connectedness corresponds to essential surjectivity on

k

-

morphisms

for

k

≤

n

+

1

.

Lemma 7.5.2 (

http://planetmath.org/75connectedness#Thmprelem1

) also implies that a type

A

is

(

-

1

)

-connected if and only if it is merely inhabited.
When a type is

0

-connected we may simply say that it is

connected

,
and when it is

1

-connected we say it is

simply connected

.

Remark 7.5.3

.

While our notion of

n

-connectedness for types agrees with the standard notion in

homotopy theory

, our notion of

n

-connectedness for

functions

is off by one from a common indexing in classical homotopy theory.
Whereas we say a function

f

is

n

-connected if all its fibers are

n

-connected, some classical

homotopy

theorists would call such a function

(

n

+

1

)

-connected.
(This is due to a historical focus on

cofibers

rather than fibers.)

We now observe a few

closure properties

of connected maps.

Lemma 7.5.4

.

Suppose that

g

is a

retract

of a

n

-connected function

f

. Then

g

is

n

-connected.

Proof.

This is a direct

consequence

of Lemma 4.7.3 (

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

).
∎

Corollary 7.5.5

.

If

g

is

homotopic

to a

n

-connected function

f

, then

g

is

n

-connected.

Lemma 7.5.6

.

Suppose that

f

:

A

→

B

is

n

-connected. Then

g

:

B

→

C

is

n

-connected if and only if

g

∘

f

is

n

-connected.

Proof.

For any

c

:

C

, we have

∥

𝖿𝗂𝖻

g

∘

f

⁢

(

c

)

∥

n

≃

∥

∑

w

:

𝖿𝗂𝖻

g

⁢

(

c

)

𝖿𝗂𝖻

f

⁢

(

𝗉𝗋

1

⁢

w

)

∥

n

(by http://planetmath.org/node/87774Exercise 4.4)

≃

∥

∑

w

:

𝖿𝗂𝖻

g

⁢

(

c

)

∥

𝖿𝗂𝖻

f

(

𝗉𝗋

1

w

)

∥

n

∥

n

(by

Theorem

7.3.9 (http://planetmath.org/73truncationshmprethm3))

≃

∥

𝖿𝗂𝖻

g

(

c

)

∥

n

.

(since $\mathopen{}\left\|{\mathsf{fib}}_{f}(\mathsf{pr}_{1}w)\right\|_{n}\mathclose{}$ is contractible)

It follows that

∥

𝖿𝗂𝖻

g

⁢

(

c

)

∥

n

is contractible if and only if

∥

𝖿𝗂𝖻

g

∘

f

⁢

(

c

)

∥

n

is
contractible.
∎

Importantly,

n

-connected functions can be equivalently characterized as those which

satisfy

an “

induction

principle” with respect to

n

-types.
This idea will lead directly into our proof of the

Freudenthal suspension theorem

in §8.6 (

http://planetmath.org/86thefreudenthalsuspensiontheorem

).

Lemma 7.5.7

.

For

f

:

A

→

B

and

P

:

B

→

U

, consider the following function:

λ

s

.

s

∘

f

:

(

∏

b

:

B

P

(

b

)

)

→

(

∏

a

:

A

P

(

f

(

a

)

)

)

.

For a fixed

f

and

n

≥

-

2

, the following are equivalent.

1.

f

is

n

-connected.

2.

For every

P

:

B

→

n

⁢

-

⁢

𝖳𝗒𝗉𝖾

, the map

λ

⁢

s

.

s

∘

f

is an

equivalence

.

3.

For every

P

:

B

→

n

⁢

-

⁢

𝖳𝗒𝗉𝖾

, the map

λ

⁢

s

.

s

∘

f

has a

section

.

Proof.

Suppose that

f

is

n

-connected and let

P

:

B

→

n

⁢

-

⁢

𝖳𝗒𝗉𝖾

. Then we have the equivalences

∏

b

:

B

P

⁢

(

b

)

≃

∏

b

:

B

(

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

→

P

(

b

)

)

(since $\mathopen{}\left\|{\mathsf{fib}}_{f}(b)\right\|_{n}\mathclose{}$ is contractible)

≃

∏

b

:

B

(

𝖿𝗂𝖻

f

(

b

)

→

P

(

b

)

)

(since $P(b)$ is an $n$-type)

≃

∏

(

b

:

B

)

∏

(

a

:

A

)

∏

(

p

:

f

(

a

)

=

b

)

P

⁢

(

b

)

(by the left

universal property

of $\Sigma$-types)

≃

∏

a

:

A

P

⁢

(

f

⁢

(

a

)

)

.

(by the left universal property of path types)

We omit the proof that this equivalence is indeed given by

λ

⁢

s

.

s

∘

f

.
Thus,

1

⇒

2

, and clearly

2

⇒

3

.
To show

3

⇒

1

, consider the type family

P

(

b

)

:

≡

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

.

Then

3

yields a map

c

:

∏

(

b

:

B

)

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

with

c

(

f

(

a

)

)

=

|

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

|

n

. To show that each

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

n

is contractible,
we will find a function of type

∏

(

b

:

B

)

∏

(

w

:

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

)

w

=

c

⁢

(

b

)

.

By Theorem 7.3.2 (

http://planetmath.org/73truncations#Thmprethm1

), for this it suffices to find a function of type

∏

(

b

:

B

)

∏

(

a

:

A

)

∏

(

p

:

f

(

a

)

=

b

)

|

(

a

,

p

)

|

n

=

c

(

b

)

.

But by rearranging

variables

and path induction, this is equivalent to the type

∏

a

:

A

|

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

|

n

=

c

(

f

(

a

)

)

.

This property holds by our choice of

c

⁢

(

f

⁢

(

a

)

)

.
∎

Corollary 7.5.8

.

For any

A

, the canonical function

|

–

|

n

:

A

→

∥

A

∥

n

is

n

-connected.

Proof.

By Theorem 7.3.2 (

http://planetmath.org/73truncations#Thmprethm1

) and the associated uniqueness principle, the condition of Lemma 7.5.7 (

http://planetmath.org/75connectedness#Thmprelem4

) holds.
∎

For

instance

, when

n

=

-

1

, Corollary 7.5.8 (

http://planetmath.org/75connectedness#Thmprecor2

) says that the map

A

→

∥

A

∥

from a type to its propositional truncation is surjective.

Corollary 7.5.9

.

A type

A

is

n

-connected if and only if the map

λ

b

.

λ

a

.

b

:

B

→

(

A

→

B

)

is an equivalence for every

n

-type

B

.
In other words, “every map from

A

to an

n

-type is

constant

”.

Proof.

By Lemma 7.5.7 (

http://planetmath.org/75connectedness#Thmprelem4

) applied to a function with codomain

𝟏

.
∎

Lemma 7.5.10

.

Let

B

be an

n

-type and let

f

:

A

→

B

be a function. Then the induced function

g

:

∥

A

∥

n

→

B

is an
equivalence if and only if

f

is

n

-connected.

Proof.

By Corollary 7.5.8 (

http://planetmath.org/75connectedness#Thmprecor2

),

|

–

|

n

is

n

-connected.
Thus, since

f

=

g

∘

|

–

|

n

, by
Lemma 7.5.6 (

http://planetmath.org/75connectedness#Thmprelem3

)

f

is

n

-connected if and only if

g

is

n

-connected.
But since

g

is a function between

n

-types, its fibers are also

n

-types.
Thus,

g

is

n

-connected if and only if it is an equivalence.
∎

We can also characterize connected pointed types in terms of connectivity of the

inclusion

of their

basepoint

.

Lemma 7.5.11

.

Let

A

be a type and

a

0

:

1

→

A

a basepoint, with

n

≥

-

1

.
Then

A

is

n

-connected if and only if the map

a

0

is

(

n

-

1

)

-connected.

Proof.

First suppose

a

0

:

𝟏

→

A

is

(

n

-

1

)

-connected and let

B

be an

n

-type; we will use Corollary 7.5.9 (

http://planetmath.org/75connectedness#Thmprecor3

).
The map

λ

b

.

λ

a

.

b

:

B

→

(

A

→

B

)

has a

retraction

given by

f

↦

f

⁢

(

a

0

)

, so it suffices to show it also has a section, i.e. that for any

f

:

A

→

B

there is

b

:

B

such that

f

=

λ

⁢

a

.

b

.
We choose

b

:

≡

f

(

a

0

)

.
Define

P

:

A

→

𝒰

by

P

(

a

)

:

≡

(

f

(

a

)

=

f

(

a

0

)

)

.
Then

P

is a family of

(

n

-

1

)

-types and we have

P

⁢

(

a

0

)

; hence we have

∏

(

a

:

A

)

P

⁢

(

a

)

since

a

0

:

𝟏

→

A

is

(

n

-

1

)

-connected.
Thus,

f

=

λ

⁢

a

.

f

⁢

(

a

0

)

as desired.

Now suppose

A

is

n

-connected, and let

P

:

A

→

(

n

-

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

and

u

:

P

⁢

(

a

0

)

be given.
By Lemma 7.5.7 (

http://planetmath.org/75connectedness#Thmprelem4

), it will suffice to

construct

f

:

∏

(

a

:

A

)

P

⁢

(

a

)

such that

f

⁢

(

a

0

)

=

u

.
Now

(

n

-

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

is an

n

-type and

A

is

n

-connected, so by Corollary 7.5.9 (

http://planetmath.org/75connectedness#Thmprecor3

), there is an

n

-type

B

such that

P

=

λ

⁢

a

.

B

.
Hence, we have a family of equivalences

g

:

∏

(

a

:

A

)

(

P

(

a

)

≃

B

)

.
Define

f

(

a

)

:

≡

g

a

-

1

(

g

a

0

(

u

)

)

; then

f

:

∏

(

a

:

A

)

P

⁢

(

a

)

and

f

⁢

(

a

0

)

=

u

as desired.
∎

In particular, a pointed type

(

A

,

a

0

)

is 0-connected if and only if

a

0

:

𝟏

→

A

is surjective, which is to say

∏

(

x

:

A

)

∥

x

=

a

0

∥

.

A useful

variation

on Lemma 7.5.6 (

http://planetmath.org/75connectedness#Thmprelem3

) is:

Lemma 7.5.12

.

Let

f

:

A

→

B

be a function and

P

:

A

→

U

and

Q

:

B

→

U

be type families. Suppose that

g

:

∏

(

a

:

A

)

P

⁢

(

a

)

→

Q

⁢

(

f

⁢

(

a

)

)

is a fiberwise

n

-connected
family of functions, i.e. each function

g

a

:

P

⁢

(

a

)

→

Q

⁢

(

f

⁢

(

a

)

)

is

n

-connected. Then the function

φ

:

(

∑

a

:

A

P

⁢

(

a

)

)

→

(

∑

b

:

B

Q

⁢

(

b

)

)

φ

⁢

(

a

,

u

)

:

≡

(

f

(

a

)

,

g

a

(

u

)

)

is

n

-connected if and only if

f

is

n

-connected.

Proof.

For

b

:

B

and

v

:

Q

⁢

(

b

)

we have

∥

𝖿𝗂𝖻

φ

(

(

b

,

v

)

)

∥

n

≃

∥

∑

(

a

:

A

)

∑

(

u

:

P

(

a

)

)

∑

(

p

:

f

(

a

)

=

b

)

f

(

p

)

*

(

g

a

(

u

)

)

=

v

∥

n

≃

∥

∑

(

w

:

𝖿𝗂𝖻

f

(

b

)

)

∑

(

u

:

P

(

𝗉𝗋

1

(

w

)

)

)

g

𝗉𝗋

1

⁢

w

(

u

)

=

f

(

𝗉𝗋

2

w

)

-

1

*

(

v

)

∥

n

≃

∥

∑

w

:

𝖿𝗂𝖻

f

⁢

(

b

)

𝖿𝗂𝖻

g

⁢

(

𝗉𝗋

1

⁢

w

)

(

f

(

𝗉𝗋

2

w

)

-

1

*

(

v

)

)

∥

n

≃

∥

∑

w

:

𝖿𝗂𝖻

f

⁢

(

b

)

∥

𝖿𝗂𝖻

g

⁢

(

𝗉𝗋

1

⁢

w

)

(

f

(

𝗉𝗋

2

w

)

-

1

*

(

v

)

)

∥

n

∥

n

≃

∥

𝖿𝗂𝖻

f

(

b

)

∥

n

where the transportations along

f

⁢

(

p

)

and

f

⁢

(

p

)

-

1

are with respect to

Q

.
Therefore, if either is contractible, so is the other.
∎

In the other direction, we have

Lemma 7.5.13

.

Let

P

,

Q

:

A

→

U

be type families and consider a fiberwise

transformation

f

:

∏

a

:

A

(

P

(

a

)

→

Q

(

a

)

)

from

P

to

Q

. Then the

induced map

total

⁢

(

f

)

:

∑

(

a

:

A

)

P

⁢

(

a

)

→

∑

(

a

:

A

)

Q

⁢

(

a

)

is

n

-connected if and only if each

f

⁢

(

a

)

is

n

-connected.

Proof.

By Theorem 4.7.6 (

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

), we have

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

(

(

x

,

v

)

)

≃

𝖿𝗂𝖻

f

⁢

(

x

)

(

v

)

for each

x

:

A

and

v

:

Q

⁢

(

x

)

. Hence

∥

𝖿𝗂𝖻

𝗍𝗈𝗍𝖺𝗅

⁢

(

f

)

(

(

x

,

v

)

)

∥

n

is contractible if and only if

∥

𝖿𝗂𝖻

f

⁢

(

x

)

⁢

(

v

)

∥

n

is contractible.
∎

Another useful fact about connected maps is that they induce an
equivalence on

n

-truncations:

Lemma 7.5.14

.

If

f

:

A

→

B

is

n

-connected, then it induces an equivalence

∥

A

∥

n

≃

∥

B

∥

n

.

Proof.

Let

c

be the proof that

f

is

n

-connected. From left to right, we
use the map

∥

f

∥

n

:

∥

A

∥

n

→

∥

B

∥

n

.
To define the map from right to left, by the universal property of
truncations, it suffices to give a map

𝖻𝖺𝖼𝗄

:

B

→

∥

A

∥

n

. We can
define this map as follows:

𝖻𝖺𝖼𝗄

(

y

)

:

≡

∥

𝗉𝗋

1

∥

n

(

𝗉𝗋

1

(

c

(

y

)

)

)

By definition,

c

⁢

(

y

)

has type

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

(

∥

𝖿𝗂𝖻

f

(

y

)

∥

n

)

, so its
first

component

has type

∥

𝖿𝗂𝖻

f

⁢

(

y

)

∥

n

, and we can obtain an
element of

∥

A

∥

n

from this by

projection

.

Next, we show that the composites are the

identity

. In both directions,
because the goal is a path in an

n

-truncated type, it suffices to
cover the case of the constructor

|

–

|

n

.

In one direction, we must show that for all

x

:

A

,

∥

𝗉𝗋

1

∥

n

(

𝗉𝗋

1

(

c

(

f

(

x

)

)

)

)

=

|

x

|

n

But

|

(

x

,

𝗋𝖾𝖿𝗅

)

|

n

:

∥

𝖿𝗂𝖻

f

(

y

)

∥

n

, and

c

⁢

(

y

)

says that this type is contractible, so

𝗉𝗋

1

(

c

(

f

(

x

)

)

)

=

|

(

x

,

𝗋𝖾𝖿𝗅

)

|

n

Applying

∥

𝗉𝗋

1

∥

n

to both sides of this

equation

gives the
result.

In the other direction, we must show that for all

y

:

B

,

∥

f

∥

n

(

∥

𝗉𝗋

1

∥

n

(

𝗉𝗋

1

(

c

(

y

)

)

)

)

=

|

y

|

n

𝗉𝗋

1

⁢

(

c

⁢

(

y

)

)

has type

∥

𝖿𝗂𝖻

f

⁢

(

y

)

∥

n

, and the path we
want is essentially the second component of the

𝖿𝗂𝖻

f

⁢

(

y

)

, but we
need to make sure the truncations work out.

In general, suppose we are given

p

:

∥

∑

(

x

:

A

)

B

(

x

)

∥

n

and wish to prove

P

(

∥

𝗉𝗋

1

∥

n

(

p

)

)

. By truncation induction, it suffices to
prove

P

(

|

a

|

n

)

for all

a

:

A

and

b

:

B

⁢

(

a

)

. Applying this
principle in this case, it suffices to prove

∥

f

∥

n

(

|

a

|

n

)

=

|

y

|

n

given

a

:

A

and

b

:

f

⁢

(

a

)

=

y

. But the left-hand side equals

|

f

⁢

(

a

)

|

n

,
so applying

|

–

|

n

to both sides of

b

gives the result.
∎

One might guess that this fact characterizes the

n

-connected maps, but in fact being

n

-connected is a bit stronger than this.
For instance, the inclusion

0

𝟐

:

𝟏

→

𝟐

induces an equivalence on

(

-

1

)

-truncations, but is not surjective (i.e.

(

-

1

)

-connected).
In §8.4 (

http://planetmath.org/84fibersequencesandthelongexactsequence

) we will see that the

difference

in general is an analogous extra bit of surjectivity.

Title

7.5 Connectedness

\metatable

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