6.4 Circles and spheres

6.4 Circles and spheres

We have already discussed the circle

𝕊

1

as the

higher inductive type

generated by

•

A point

𝖻𝖺𝗌𝖾

:

𝕊

1

, and

•

A path

𝗅𝗈𝗈𝗉

:

𝖻𝖺𝗌𝖾

=

𝕊

1

𝖻𝖺𝗌𝖾

.

Its

induction

principle says that given

P

:

𝕊

1

→

𝒰

along with

b

:

P

⁢

(

𝖻𝖺𝗌𝖾

)

and

ℓ

:

b

=

𝗅𝗈𝗈𝗉

P

b

, we have

f

:

∏

(

x

:

𝕊

1

)

P

⁢

(

x

)

with

f

⁢

(

𝖻𝖺𝗌𝖾

)

≡

b

and

𝖺𝗉𝖽

f

(

𝗅𝗈𝗈𝗉

)

=

ℓ

.
Its non-dependent

recursion

principle says that given

B

with

b

:

B

and

ℓ

:

b

=

b

, we have

f

:

𝕊

1

→

B

with

f

⁢

(

𝖻𝖺𝗌𝖾

)

≡

b

and

f

(

𝗅𝗈𝗈𝗉

)

=

ℓ

.

We observe that the circle is nontrivial.

Lemma 6.4.1

.

𝗅𝗈𝗈𝗉

≠

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

.

Proof.

Suppose that

𝗅𝗈𝗈𝗉

=

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

.
Then since for any type

A

with

x

:

A

and

p

:

x

=

x

, there is a

function

f

:

𝕊

1

→

A

defined by

f

(

𝖻𝖺𝗌𝖾

)

:

≡

x

and

f

(

𝗅𝗈𝗈𝗉

)

:=

p

, we have

p

=

f

⁢

(

𝗅𝗈𝗈𝗉

)

=

f

⁢

(

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

)

=

𝗋𝖾𝖿𝗅

x

.

But this implies that every type is a set, which as we have seen is not the case (see Example 3.1.9 (

http://planetmath.org/31setsandntypes#Thmpreeg6

)).
∎

The circle also has the following interesting property, which is useful as a source of counterexamples.

Lemma 6.4.2

.

There exists an element of

∏

(

x

:

S

1

)

(

x

=

x

)

which is not equal to

x

↦

refl

x

.

Proof.

We define

f

:

∏

(

x

:

𝕊

1

)

(

x

=

x

)

by

𝕊

1

-induction.
When

x

is

𝖻𝖺𝗌𝖾

, we let

f

(

𝖻𝖺𝗌𝖾

)

:

≡

𝗅𝗈𝗈𝗉

.
Now when

x

varies along

𝗅𝗈𝗈𝗉

(see Remark 6.2.4 (

http://planetmath.org/62inductionprinciplesanddependentpaths#Thmprermk3

)), we must show that

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

x

=

x

⁢

(

𝗅𝗈𝗈𝗉

,

𝗅𝗈𝗈𝗉

)

=

𝗅𝗈𝗈𝗉

.
However, in §2.11 (

http://planetmath.org/211identitytype

) we observed that

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

x

=

x

⁢

(

p

,

q

)

=

p

-

1

⁢

\centerdot

⁢

q

⁢

\centerdot

⁢

p

, so what we have to show is that

𝗅𝗈𝗈𝗉

-

1

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

=

𝗅𝗈𝗈𝗉

.
But this is clear by canceling an

inverse

.

To show that

f

≠

(

x

↦

𝗋𝖾𝖿𝗅

x

)

, it suffices by function extensionality to show that

f

⁢

(

𝖻𝖺𝗌𝖾

)

≠

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

.
But

f

⁢

(

𝖻𝖺𝗌𝖾

)

=

𝗅𝗈𝗈𝗉

, so this is just the previous lemma.
∎

For

instance

, this enables us to extend Example 3.1.9 (

http://planetmath.org/31setsandntypes#Thmpreeg6

) by showing that any

universe

which contains the circle cannot be a 1-type.

Corollary 6.4.3

.

If the type

S

1

belongs to some universe

U

, then

U

is not a 1-type.

Proof.

The type

𝕊

1

=

𝕊

1

in

𝒰

is, by univalence,

equivalent

to the type

𝕊

1

≃

𝕊

1

of autoequivalences of

𝕊

1

, so it suffices to show that

𝕊

1

≃

𝕊

1

is not a set.
For this, it suffices to show that its equality type

𝗂𝖽

𝕊

1

=

(

𝕊

1

≃

𝕊

1

)

𝗂𝖽

𝕊

1

is not a mere proposition.
Since being an equivalence is a mere proposition, this type is equivalent to

𝗂𝖽

𝕊

1

=

(

𝕊

1

→

𝕊

1

)

𝗂𝖽

𝕊

1

.
But by function extensionality, this is equivalent to

∏

(

x

:

𝕊

1

)

(

x

=

x

)

, which as we have seen in Lemma 6.4.2 (

http://planetmath.org/64circlesandspheres#Thmprelem2

) contains two unequal

elements

.
∎

We have also mentioned that the 2-sphere

𝕊

2

should be the higher inductive type generated by

•

A point

𝖻𝖺𝗌𝖾

:

𝕊

2

, and

•

A 2-dimensional path

𝗌𝗎𝗋𝖿

:

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

=

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

in

𝖻𝖺𝗌𝖾

=

𝖻𝖺𝗌𝖾

.

The recursion principle for

𝕊

2

is not hard: it says that given

B

with

b

:

B

and

s

:

𝗋𝖾𝖿𝗅

b

=

𝗋𝖾𝖿𝗅

b

, we have

f

:

𝕊

2

→

B

with

f

⁢

(

𝖻𝖺𝗌𝖾

)

≡

b

and

𝖺𝗉

f

2

(

𝗌𝗎𝗋𝖿

)

=

s

.
Here by “

𝖺𝗉

f

2

(

𝗌𝗎𝗋𝖿

)

” we mean an

extension

of the functorial action of

f

to two-dimensional paths, which can be stated precisely as follows.

Lemma 6.4.4

.

Given

f

:

A

→

B

and

x

,

y

:

A

and

p

,

q

:

x

=

y

, and

r

:

p

=

q

, we have a path

ap

f

2

(

r

)

:

f

(

p

)

=

f

(

q

)

.

Proof.

By path induction, we may assume

p

≡

q

and

r

is

reflexivity

.
But then we may define

𝖺𝗉

f

2

(

𝗋𝖾𝖿𝗅

p

)

:

≡

𝗋𝖾𝖿𝗅

f

(

p

)

.
∎

In order to state the general induction principle, we need a version of this lemma for dependent functions, which in turn requires a notion of dependent two-dimensional paths.
As before, there are many ways to define such a thing; one is by way of a two-dimensional version of transport.

Lemma 6.4.5

.

Given

P

:

A

→

U

and

x

,

y

:

A

and

p

,

q

:

x

=

y

and

r

:

p

=

q

, for any

u

:

P

⁢

(

x

)

we have

transport

2

(

r

,

u

)

:

p

*

(

u

)

=

q

*

(

u

)

.

Proof.

By path induction.
∎

Now suppose given

x

,

y

:

A

and

p

,

q

:

x

=

y

and

r

:

p

=

q

and also points

u

:

P

⁢

(

x

)

and

v

:

P

⁢

(

y

)

and dependent paths

h

:

u

=

p

P

v

and

k

:

u

=

q

P

v

.
By our definition of dependent paths, this means

h

:

p

*

(

u

)

=

v

and

k

:

q

*

(

u

)

=

v

.
Thus, it is reasonable to define the type of dependent 2-paths over

r

to be

(

h

=

r

P

k

)

:

≡

(

h

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

2

(

r

,

u

)

\centerdot

k

)

.

We can now state the dependent version of Lemma 6.4.4 (

http://planetmath.org/64circlesandspheres#Thmprelem3

).

Lemma 6.4.6

.

Given

P

:

A

→

U

and

x

,

y

:

A

and

p

,

q

:

x

=

y

and

r

:

p

=

q

and a function

f

:

∏

(

x

:

A

)

P

⁢

(

x

)

, we have

apd

f

2

(

r

)

:

apd

f

(

p

)

=

r

P

apd

f

(

q

)

.

Proof.

Path induction.
∎

Now we can state the induction principle for

𝕊

2

: given

P

:

𝕊

2

→

P

with

b

:

P

⁢

(

𝖻𝖺𝗌𝖾

)

and

s

:

𝗋𝖾𝖿𝗅

b

=

𝗌𝗎𝗋𝖿

P

𝗋𝖾𝖿𝗅

b

, there is a function

f

:

∏

(

x

:

𝕊

2

)

P

⁢

(

x

)

such that

f

⁢

(

𝖻𝖺𝗌𝖾

)

≡

b

and

𝖺𝗉𝖽

f

2

(

𝗌𝗎𝗋𝖿

)

=

s

.

Of course, this explicit approach gets more and more complicated as we go up in

dimension

.
Thus, if we want to define

n

-spheres for all

n

, we need some more systematic idea.
One approach is to work with

n

-dimensional loops directly, rather than general

n

-dimensional paths.

Recall from §2.1 (

http://planetmath.org/21typesarehighergroupoids

) the definitions of

pointed types

𝒰

*

, and the

n

-fold

loop space

Ω

n

:

𝒰

*

→

𝒰

*

(Definition 2.1.7 (

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

) and Definition 2.1.8 (

http://planetmath.org/21typesarehighergroupoids#Thmpredefn2

)). Now we can define the

n

-sphere

𝕊

n

to be the higher inductive type generated by

•

A point

𝖻𝖺𝗌𝖾

:

𝕊

n

, and

•

An

n

-loop

𝗅𝗈𝗈𝗉

n

:

Ω

n

⁢

(

𝕊

n

,

𝖻𝖺𝗌𝖾

)

.

In order to write down the induction principle for this

presentation

, we would need to define a notion of “dependent

n

-loop”, along with the action of dependent functions on

n

-loops.
We leave this to the reader (see

http://planetmath.org/node/87769

Exercise 6.4); in the next

section

we will discuss a different way to define the spheres that is sometimes more tractable.

Title

6.4 Circles and spheres

\metatable

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