8.9 A general statement of the encode-decode method

8.9 A general statement of the encode-decode method

We have used the encode-decode method to characterize the

path spaces

of various types, including

coproducts

(

\cref

thm:path-coprod),

natural
numbers

(

\cref

thm:path-nat), truncations (

\cref

thm:path-truncation),
the circle (

\cref

cor:omega-s1),

suspensions

(

\autoref

thm:freudenthal), and

pushouts

(

\cref

thm:van-Kampen). Variants of this technique are used in the
proofs of many of the other

theorems

mentioned in the

introduction

to
this chapter, such as a direct proof of

π

n

⁢

(

𝕊

n

)

, the Blakers–Massey theorem, and the construction of Eilenberg–Mac Lane spaces.
While it is tempting to try to
abstract the method into a lemma, this is difficult because
slightly different variants are needed for different problems. For
example, different variations on the same method can be used to
characterize a

loop space

(as in

\cref

thm:path-coprod,cor:omega-s1) or
a whole path space (as in

\cref

thm:path-nat), to give a

complete

characterization

of a loop space (e.g.

Ω

1

⁢

(

𝕊

1

)

) or only to
characterize some truncation of it (e.g. van Kampen), and to calculate

homotopy groups

or to prove that a map is

n

-

connected

(e.g. Freudenthal and
Blakers–Massey).

However, we can state lemmas for specific variants of the method.
The proofs of these lemmas are almost

trivial

; the main point is to
clarify the method by stating them in generality. The simplest
case is using an encode-decode method to characterize the loop space of a
type, as in

\cref

thm:path-coprod and

\cref

cor:omega-s1.

Lemma 8.9.1

(Encode-decode for Loop Spaces).

Given a pointed type

(

A

,

a

0

)

and a

fibration

code

:

A

→

U

, if

1.

c

0

:

𝖼𝗈𝖽𝖾

⁢

(

a

0

)

,

2.

𝖽𝖾𝖼𝗈𝖽𝖾

:

∏

(

x

:

A

)

𝖼𝗈𝖽𝖾

(

x

)

→

(

a

0

=

x

)

,

3.

for all

c

:

𝖼𝗈𝖽𝖾

⁢

(

a

0

)

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

⁢

(

𝖽𝖾𝖼𝗈𝖽𝖾

⁢

(

c

)

,

c

0

)

=

c

, and

4.

𝖽𝖾𝖼𝗈𝖽𝖾

⁢

(

c

0

)

=

𝗋𝖾𝖿𝗅

,

then

(

a

0

=

a

0

)

is

equivalent

to

code

⁢

(

a

0

)

.

Proof.

Define

𝖾𝗇𝖼𝗈𝖽𝖾

:

∏

(

x

:

A

)

(

a

0

=

x

)

→

𝖼𝗈𝖽𝖾

(

x

)

by

𝖾𝗇𝖼𝗈𝖽𝖾

x

⁢

(

α

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

⁢

(

α

,

c

0

)

.

We show that

𝖾𝗇𝖼𝗈𝖽𝖾

a

0

and

𝖽𝖾𝖼𝗈𝖽𝖾

a

0

are

quasi-inverses

.
The

composition

𝖾𝗇𝖼𝗈𝖽𝖾

a

0

∘

𝖽𝖾𝖼𝗈𝖽𝖾

a

0

is immediate by

assumption

3

. For the other composition, we show

∏

(

x

:

A

)

∏

(

p

:

a

0

=

x

)

𝖽𝖾𝖼𝗈𝖽𝖾

x

⁢

(

𝖾𝗇𝖼𝗈𝖽𝖾

x

⁢

p

)

=

p

.

By path

induction

, it suffices to show

𝖽𝖾𝖼𝗈𝖽𝖾

a

0

⁢

(

𝖾𝗇𝖼𝗈𝖽𝖾

a

o

⁢

𝗋𝖾𝖿𝗅

)

=

𝗋𝖾𝖿𝗅

.
After reducing the

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

, it suffices to show

𝖽𝖾𝖼𝗈𝖽𝖾

a

0

⁢

(

c

0

)

=

𝗋𝖾𝖿𝗅

, which is assumption

4

.
∎

If a fiberwise

equivalence

between

(

a

0

=

–

)

and

𝖼𝗈𝖽𝖾

is desired,
it suffices to strengthen condition (iii) to

∏

(

x

:

A

)

∏

(

c

:

𝖼𝗈𝖽𝖾

(

x

)

)

𝖾𝗇𝖼𝗈𝖽𝖾

x

⁢

(

𝖽𝖾𝖼𝗈𝖽𝖾

x

⁢

(

c

)

)

=

c

.

a

However, to calculate a loop space (e.g.

Ω

⁢

(

𝕊

1

)

), this

stronger

assumption is not necessary.

Another variation, which comes up often when calculating homotopy
groups, characterizes the truncation of a loop space:

Lemma 8.9.2

(Encode-decode for Truncations of Loop Spaces).

Assume a pointed type

(

A

,

a

0

)

and a fibration

code

:

A

→

U

, where for every

x

,

code

⁢

(

x

)

is a

k

-type.
Define

𝖾𝗇𝖼𝗈𝖽𝖾

:

∏

x

:

A

∥

a

0

=

x

∥

k

→

𝖼𝗈𝖽𝖾

(

x

)

.

by truncation

recursion

(using the fact
that

code

⁢

(

x

)

is a

k

-type),

mapping

α

:

a

0

=

x

to

transport

code

⁢

(

α

,

c

0

)

. Suppose:

1.

c

0

:

𝖼𝗈𝖽𝖾

⁢

(

a

0

)

,

2.

𝖽𝖾𝖼𝗈𝖽𝖾

:

∏

(

x

:

A

)

𝖼𝗈𝖽𝖾

(

x

)

→

∥

a

0

=

x

∥

k

,

3.

𝖾𝗇𝖼𝗈𝖽𝖾

a

0

⁢

(

𝖽𝖾𝖼𝗈𝖽𝖾

a

⁢

0

⁢

(

c

)

)

=

c

for all

c

:

𝖼𝗈𝖽𝖾

⁢

(

a

0

)

, and

4.

𝖽𝖾𝖼𝗈𝖽𝖾

(

c

0

)

=

|

𝗋𝖾𝖿𝗅

|

.

Then

∥

a

0

=

a

0

∥

k

is equivalent to

code

⁢

(

a

0

)

.

Proof.

That

𝖽𝖾𝖼𝗈𝖽𝖾

∘

𝖾𝗇𝖼𝗈𝖽𝖾

is

identity

is immediate by

3

.
To prove

𝖾𝗇𝖼𝗈𝖽𝖾

∘

𝖽𝖾𝖼𝗈𝖽𝖾

, we first do a truncation induction, by
which it suffices to show

∏

(

x

:

A

)

∏

(

p

:

a

0

=

x

)

𝖽𝖾𝖼𝗈𝖽𝖾

x

(

𝖾𝗇𝖼𝗈𝖽𝖾

x

(

|

p

|

k

)

)

=

|

p

|

k

.

The truncation induction is allowed because paths in a

k

-type are a

k

-type. To show this type, we do a path induction, and after reducing
the

𝖾𝗇𝖼𝗈𝖽𝖾

, use assumption

4

.
∎

Title

8.9 A general statement of the encode-decode method

\metatable

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