8.1.6 The universal cover as an identity system

8.1.6 The universal cover as an identity system

Note that the fibration

𝖼𝗈𝖽𝖾

:

𝕊

1

→

𝒰

together with

0

:

𝖼𝗈𝖽𝖾

⁢

(

𝖻𝖺𝗌𝖾

)

is a

pointed

predicate

in the sense of

\autoref

defn:identity-systems.
From this point of view, we can see that the encode-decode proof in

\autoref

subsec:pi1s1-encode-decode consists of proving that

𝖼𝗈𝖽𝖾

satisfies

\autoref

thm:identity-systems

LABEL:item:identity-systems3

, while the homotopy-theoretic proof in

\autoref

subsec:pi1s1-homotopy-theory consists of proving that it satisfies

\autoref

thm:identity-systems

LABEL:item:identity-systems4

.
This suggests a third approach.

Theorem 8.1.1

.

The pair

(

code

,

0

)

is an

identity

system at

base

:

S

1

in the sense of

\autoref

defn:identity-systems.

Proof.

Let

D

:

∏

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

⁢

(

x

)

→

𝒰

and

d

:

D

⁢

(

𝖻𝖺𝗌𝖾

,

0

)

be given; we want to define a

function

f

:

∏

(

x

:

𝕊

1

)

∏

(

c

:

𝖼𝗈𝖽𝖾

(

x

)

)

D

⁢

(

x

,

c

)

.
By circle

induction

, it suffices to specify

f

⁢

(

𝖻𝖺𝗌𝖾

)

:

∏

(

c

:

𝖼𝗈𝖽𝖾

(

𝖻𝖺𝗌𝖾

)

)

D

⁢

(

𝖻𝖺𝗌𝖾

,

c

)

and verify that

𝗅𝗈𝗈𝗉

*

(

f

(

𝖻𝖺𝗌𝖾

)

)

=

f

(

𝖻𝖺𝗌𝖾

)

.

Of course,

𝖼𝗈𝖽𝖾

⁢

(

𝖻𝖺𝗌𝖾

)

≡

ℤ

.
By

\autoref

lem:transport-s1-code and induction on

n

, we may obtain a path

p

n

:

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

⁢

(

𝗅𝗈𝗈𝗉

n

,

0

)

=

n

for any integer

n

.
Therefore, by paths in

Σ

-types, we have a path

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

n

,

p

n

)

:

(

𝖻𝖺𝗌𝖾

,

0

)

=

(

𝖻𝖺𝗌𝖾

,

n

)

in

∑

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

⁢

(

x

)

.
Transporting

d

along this path in the fibration

D

^

:

(

∑

(

x

:

𝕊

1

)

𝖼𝗈𝖽𝖾

⁢

(

x

)

)

→

𝒰

associated to

D

, we obtain an element of

D

⁢

(

𝖻𝖺𝗌𝖾

,

n

)

for any

n

:

ℤ

.
We define this

element

to be

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

)

:

f

(

𝖻𝖺𝗌𝖾

)

(

n

)

:

≡

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

(

𝗉𝖺𝗂𝗋

=

(

𝗅𝗈𝗈𝗉

n

,

p

n

)

,

d

)

.

Now we need

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

λ

⁢

x

.

∏

(

c

:

𝖼𝗈𝖽𝖾

(

x

)

)

D

⁢

(

x

,

c

)

⁢

(

𝗅𝗈𝗈𝗉

,

f

⁢

(

𝖻𝖺𝗌𝖾

)

)

=

f

⁢

(

𝖻𝖺𝗌𝖾

)

.
By

\autoref

thm:dpath-forall, this means we need to show that for any

n

:

ℤ

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

(

𝗉𝖺𝗂𝗋

=

(

𝗅𝗈𝗈𝗉

,

𝗋𝖾𝖿𝗅

𝗅𝗈𝗈𝗉

*

(

n

)

)

,

f

(

𝖻𝖺𝗌𝖾

)

(

n

)

)

=

D

(

𝖻𝖺𝗌𝖾

,

𝗅𝗈𝗈𝗉

*

(

n

)

)

f

(

𝖻𝖺𝗌𝖾

)

(

𝗅𝗈𝗈𝗉

*

(

n

)

)

.

Now we have a path

q

:

𝗅𝗈𝗈𝗉

*

(

n

)

=

n

+

1

, so transporting along this, it suffices to show

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

q

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

,

𝗋𝖾𝖿𝗅

𝗅𝗈𝗈𝗉

*

(

n

)

)

,

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

)

)

)

=

D

⁢

(

𝖻𝖺𝗌𝖾

,

n

+

1

)

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

⁢

(

𝖻𝖺𝗌𝖾

)

(

q

,

f

(

𝖻𝖺𝗌𝖾

)

(

𝗅𝗈𝗈𝗉

*

(

n

)

)

)

.

By a couple of lemmas about transport and dependent application, this is

equivalent

to

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

,

q

)

,

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

)

)

=

D

⁢

(

𝖻𝖺𝗌𝖾

,

n

+

1

)

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

+

1

)

.

However, expanding out the

definition

of

f

⁢

(

𝖻𝖺𝗌𝖾

)

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

,

q

)

,

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

)

)

⁢

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

,

q

)

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

n

,

p

n

)

,

d

)

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

n

,

p

n

)

⁢

\centerdot

⁢

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

,

q

)

,

d

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

D

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗅𝗈𝗈𝗉

n

+

1

,

p

n

+

1

)

,

d

)

=

f

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

(

n

+

1

)

.

We have used the functoriality of transport, the

characterization

of composition in

Σ

-types (which was an exercise for the reader), and a lemma relating

p

n

and

q

to

p

n

+

1

which we leave it to the reader to state and prove.

This

completes

the construction of

f

:

∏

(

x

:

𝕊

1

)

∏

(

c

:

𝖼𝗈𝖽𝖾

(

x

)

)

D

⁢

(

x

,

c

)

.
Since

f

(

𝖻𝖺𝗌𝖾

,

0

)

≡

𝗉𝖺𝗂𝗋

=

(

𝗅𝗈𝗈𝗉

0

,

p

0

)

*

(

d

)

=

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

*

(

d

)

=

d

,

we have shown that

(

𝖼𝗈𝖽𝖾

,

0

)

is an identity system.
∎

Corollary 8.1.2

.

For any

x

:

S

1

, we have

(

base

=

x

)

≃

code

(

x

)

.

Proof.

By

\autoref

thm:identity-systems.
∎

Of course, this proof also contains essentially the same elements as the previous two.
Roughly, we can say that it unifies the proofs of

\autoref

thm:pi1s1-decode,

\autoref

lem:s1-encode-decode, performing the requisite inductive

argument

only once in a

generic

case.

Title

8.1.6 The

universal cover

as an identity system

\metatable

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