8.1.5 The homotopy-theoretic proof

8.1.5 The homotopy-theoretic proof

In

\autoref

sec:pi1s1-universal-cover, we defined the putative

universal cover

𝖼𝗈𝖽𝖾

:

𝕊

1

→

𝒰

in

type theory

, and in

\autoref

subsec:pi1s1-homotopy-theory we defined a map

𝖾𝗇𝖼𝗈𝖽𝖾

:

∏

(

x

:

𝕊

1

)

(

𝖻𝖺𝗌𝖾

=

x

)

→

𝖼𝗈𝖽𝖾

(

x

)

from the path

fibration

to the universal cover.
What remains for the classical proof is to show that this map induces an

equivalence

on

total spaces

because both are

contractible

, and to deduce from this that it must be an equivalence on each fiber.

In

\autoref

thm:contr-paths we saw that the total space

∑

(

x

:

𝕊

1

)

(

𝖻𝖺𝗌𝖾

=

x

)

is contractible.
For the other, we have:

Lemma 8.1.1

.

The type

∑

(

x

:

S

1

)

code

⁢

(

x

)

is contractible.

Proof.

We apply the flattening lemma (

\autoref

thm:flattening) with the following values:

•

A

:

≡

𝟏

and

B

:

≡

𝟏

, with

f

and

g

the

obvious

functions

.
Thus, the base

higher inductive type

W

in the flattening lemma is

equivalent

to

𝕊

1

.

•

C

:

A

→

𝒰

is

constant

at

ℤ

.

•

D

:

∏

(

b

:

B

)

(

ℤ

≃

ℤ

)

is constant at

𝗌𝗎𝖼𝖼

.

Then the type family

P

:

𝕊

1

→

𝒰

defined in the flattening lemma is equivalent to

𝖼𝗈𝖽𝖾

:

𝕊

1

→

𝒰

.
Thus, the flattening lemma tells us that

∑

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

⁢

(

x

)

is equivalent to a higher inductive type with the following

generators

, which we denote

R

:

•

A function

𝖼

:

ℤ

→

R

.

•

For each

z

:

ℤ

, a path

𝗉

z

:

𝖼

⁢

(

z

)

=

𝖼

⁢

(

𝗌𝗎𝖼𝖼

⁢

(

z

)

)

.

We might call this type the

homotopical reals

;
it plays the same role as the

topological space

ℝ

in the classical proof.

Thus, it remains to show that

R

is contractible.
As center of

contraction

we choose

𝖼

⁢

(

0

)

; we must now show that

x

=

𝖼

⁢

(

0

)

for all

x

:

R

.
We do this by

induction

on

R

.
Firstly, when

x

is

𝖼

⁢

(

z

)

, we must give a path

q

z

:

𝖼

⁢

(

0

)

=

𝖼

⁢

(

z

)

, which we can do by induction on

z

:

ℤ

:

q

0

:

≡

𝗋𝖾𝖿𝗅

𝖼

⁢

(

0

)

q

n

+

1

:

≡

q

n

\centerdot

𝗉

n

for

n

≥

0

q

n

-

1

:

≡

q

n

\centerdot

𝗉

n

-

1

-

1

for

n

≤

0

.

Secondly, we must show that for any

z

:

ℤ

, the path

q

z

is transported along

𝗉

z

to

q

z

+

1

.
By transport of paths, this means we want

q

z

⁢

\centerdot

⁢

𝗉

z

=

q

z

+

1

.
This is easy by induction on

z

, using the

definition

of

q

z

.
This

completes

the proof that

R

is contractible, and thus so is

∑

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

⁢

(

x

)

.
∎

Corollary 8.1.2

.

The map induced by

encode

:

∑

(

x

:

𝕊

1

)

(

𝖻𝖺𝗌𝖾

=

x

)

→

∑

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

(

x

)

is an equivalence.

Proof.

Both types are contractible.
∎

Theorem 8.1.3

.

Ω

⁢

(

𝕊

1

,

𝖻𝖺𝗌𝖾

)

≃

ℤ

.

Proof.

Apply

\autoref

thm:total-fiber-equiv to

𝖾𝗇𝖼𝗈𝖽𝖾

, using

\autoref

thm:encode-total-equiv.
∎

In essence, the two proofs are not very different: the encode-decode one may be seen as a “

reduction

” or “unpackaging” of the homotopy-theoretic one.
Each has its advantages; the interplay between the two points of view is part of the

interest

of the subject.

Title

8.1.5 The homotopy-theoretic proof

\metatable

Generated on Thu Feb 8 19:40:33 2018 by

LaTeXML