4.2 Half adjoint equivalences

4.2 Half adjoint equivalences

In §4.1 (

http://planetmath.org/41quasiinverses

) we concluded that

𝗊𝗂𝗇𝗏

⁢

(

f

)

is

equivalent

to

∏

(

x

:

A

)

(

x

=

x

)

by discarding a contractible type.
Roughly, the type

𝗊𝗂𝗇𝗏

⁢

(

f

)

contains three data

g

,

η

, and

ϵ

, of which two (

g

and

η

) could together be seen to be

contractible

when

f

is an

equivalence

.
The problem is that removing these data left one remaining (

ϵ

).
In order to solve this problem, the idea is to add one

additional

datum which, together with

ϵ

, forms a contractible type.

Definition 4.2.1

.

A

function

f

:

A

→

B

is a

half adjoint equivalence

if there are

g

:

B

→

A

and

homotopies

η

:

g

∘

f

∼

id

A

and

ϵ

:

f

∘

g

∼

id

B

such that there exists a homotopy

τ

:

∏

x

:

A

f

(

η

x

)

=

ϵ

(

f

x

)

.

Thus we have a type

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

, defined to be

∑

(

g

:

B

→

A

)

∑

(

η

:

g

∘

f

∼

𝗂𝖽

A

)

∑

(

ϵ

:

f

∘

g

∼

𝗂𝖽

B

)

∏

(

x

:

A

)

f

(

η

x

)

=

ϵ

(

f

x

)

.

Note that in the above

definition

, the coherence condition relating

η

and

ϵ

only involves

f

.
We might consider instead an analogous coherence condition involving

g

:

υ

:

∏

y

:

B

g

(

ϵ

y

)

=

η

(

g

y

)

and a resulting analogous definition

𝗂𝗌𝗁𝖺𝖾

′

⁢

(

f

)

.

Fortunately, it turns out each of the conditions implies the other one:

Lemma 4.2.2

.

For functions

f

:

A

→

B

and

g

:

B

→

A

and homotopies

η

:

g

∘

f

∼

id

A

and

ϵ

:

f

∘

g

∼

id

B

, the following conditions are

logically equivalent

:

•

∏

(

x

:

A

)

f

(

η

x

)

=

ϵ

(

f

x

)

•

∏

(

y

:

B

)

g

(

ϵ

y

)

=

η

(

g

y

)

Proof.

It suffices to show one direction; the other one is obtained by replacing

A

,

f

, and

η

by

B

,

g

, and

ϵ

respectively.
Let

τ

:

∏

(

x

:

A

)

f

(

η

x

)

=

ϵ

(

f

x

)

.
Fix

y

:

B

.
Using naturality of

ϵ

and applying

g

, we get the following commuting

diagram

of paths:

\includegraphics

HoTT_fig_4.2.1.png

Using

τ

⁢

(

g

⁢

y

)

on the left side of the diagram gives us

\includegraphics

HoTT_fig_4.2.2.png

Using the

commutativity

of

η

with

g

∘

f

(Definition 2 (

http://planetmath.org/24homotopiesandequivalences#Thmdefn2

)), we have

\includegraphics

HoTT_fig_4.2.3.png

However, by naturality of

η

we also have

\includegraphics

HoTT_fig_4.2.4.png

Thus, canceling all but the right-hand homotopy, we have

g

⁢

(

ϵ

⁢

y

)

=

η

⁢

(

g

⁢

y

)

as desired.
∎

However, it is important that we do not include

both

τ

and

υ

in the definition of

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

(whence the name “

half

adjoint equivalence”).
If we did, then after canceling contractible types we would still have one remaining datum — unless we added another higher coherence condition.
In general, we expect to get a well-behaved type if we cut off after an

odd number

of coherences.

Of course, it is

obvious

that

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

: simply forget the coherence datum.
The other direction is a version of a standard

argument

from

homotopy theory

and

category theory

.

Theorem 4.2.3

.

For any

f

:

A

→

B

we have

qinv

⁢

(

f

)

→

ishae

⁢

(

f

)

.

Proof.

Suppose that

(

g

,

η

,

ϵ

)

is a

quasi-inverse

for

f

. We have to provide
a quadruple

(

g

′

,

η

′

,

ϵ

′

,

τ

)

witnessing that

f

is a half adjoint equivalence. To
define

g

′

and

η

′

, we can just make the obvious choice by setting

g

′

:

≡

g

and

η

′

:

≡

η

. However, in the definition of

ϵ

′

we
need start worrying about the construction of

τ

, so we cannot just follow our nose
and take

ϵ

′

to be

ϵ

. Instead, we take

ϵ

′

(

b

)

:

≡

ϵ

⁢

(

f

⁢

(

g

⁢

(

b

)

)

)

-

1

\centerdot

(

f

(

η

(

g

(

b

)

)

)

\centerdot

ϵ

(

b

)

)

.

Now we need to find

τ

(

a

)

:

ϵ

⁢

(

f

⁢

(

g

⁢

(

f

⁢

(

a

)

)

)

)

-

1

\centerdot

(

f

(

η

(

g

(

f

(

a

)

)

)

)

\centerdot

ϵ

(

f

(

a

)

)

)

=

f

(

η

(

a

)

)

.

Note first that by

Corollary

2.4.4 (

http://planetmath.org/24homotopiesandequivalences#Thmprelem2

), we have

η

(

g

(

f

(

a

)

)

)

=

g

(

f

(

η

(

a

)

)

)

. Therefore, we can apply
Lemma 2.4.3 (

http://planetmath.org/24homotopiesandequivalences#Thmcor1

) to compute

f

(

η

(

g

(

f

(

a

)

)

)

)

\centerdot

ϵ

(

f

(

a

)

)

=

f

(

g

(

f

(

η

(

a

)

)

)

)

\centerdot

ϵ

(

f

(

a

)

)

=

ϵ

(

f

(

g

(

f

(

a

)

)

)

)

\centerdot

f

(

η

(

a

)

)

from which we get the desired path

τ

⁢

(

a

)

.
∎

Combining this with Lemma 4.2.2 (

http://planetmath.org/42halfadjointequivalences#Thmprelem1

) (or symmetrizing the proof), we also have

𝗊𝗂𝗇𝗏

⁢

(

f

)

→

𝗂𝗌𝗁𝖺𝖾

′

⁢

(

f

)

.

It remains to show that

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

is a mere proposition.
For this, we will need to know that the fibers of an equivalence are contractible.

Definition 4.2.4

.

The

fiber

of a map

f

:

A

→

B

over a point

y

:

B

is

𝖿𝗂𝖻

f

(

y

)

:

≡

∑

x

:

A

(

f

(

x

)

=

y

)

.

In homotopy theory, this is what would be called the

homotopy fiber

of

f

.
The path lemmas in §2.5 (

http://planetmath.org/25thehighergroupoidstructureoftypeformers

) yield the following

characterization

of paths in fibers:

Lemma 4.2.5

.

For any

f

:

A

→

B

,

y

:

B

, and

(

x

,

p

)

,

(

x

′

,

p

′

)

:

fib

f

⁢

(

y

)

, we have

(

(

x

,

p

)

=

(

x

′

,

p

′

)

)

≃

(

∑

γ

:

x

=

x

′

f

(

γ

)

\centerdot

p

′

=

p

)

Theorem 4.2.6

.

If

f

:

A

→

B

is a half adjoint equivalence, then for any

y

:

B

the fiber

fib

f

⁢

(

y

)

is contractible.

Proof.

Let

(

g

,

η

,

ϵ

,

τ

)

:

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

, and fix

y

:

B

.
As our center of

contraction

for

𝖿𝗂𝖻

f

⁢

(

y

)

we choose

(

g

⁢

y

,

ϵ

⁢

y

)

.
Now take any

(

x

,

p

)

:

𝖿𝗂𝖻

f

⁢

(

y

)

; we want to

construct

a path from

(

g

⁢

y

,

ϵ

⁢

y

)

to

(

x

,

p

)

.
By Lemma 4.2.5 (

http://planetmath.org/42halfadjointequivalences#Thmprelem2

), it suffices to give a path

γ

:

g

⁢

y

=

x

such that

f

(

γ

)

\centerdot

p

=

ϵ

y

.
We put

γ

:

≡

g

⁢

(

p

)

-

1

\centerdot

η

x

.
Then we have

f

⁢

(

γ

)

⁢

\centerdot

⁢

p

=

f

⁢

g

⁢

(

p

)

-

1

⁢

\centerdot

⁢

f

⁢

(

η

⁢

x

)

⁢

\centerdot

⁢

p

=

f

⁢

g

⁢

(

p

)

-

1

⁢

\centerdot

⁢

ϵ

⁢

(

f

⁢

x

)

⁢

\centerdot

⁢

p

=

ϵ

⁢

y

where the second equality follows by

τ

⁢

x

and the third equality is naturality of

ϵ

.
∎

We now define the types which encapsulate contractible pairs of data.
The following types put together the quasi-inverse

g

with one of the homotopies.

Definition 4.2.7

.

Given a function

f

:

A

→

B

, we define the types

𝗅𝗂𝗇𝗏

⁢

(

f

)

:

≡

∑

g

:

B

→

A

(

g

∘

f

∼

𝗂𝖽

A

)

𝗋𝗂𝗇𝗏

⁢

(

f

)

:

≡

∑

g

:

B

→

A

(

f

∘

g

∼

𝗂𝖽

B

)

of

left inverses

and

right inverses

to

f

, respectively.
We call

f

left invertible

if

linv

⁢

(

f

)

is inhabited, and similarly

right invertible

if

rinv

⁢

(

f

)

is inhabited.

Lemma 4.2.8

.

If

f

:

A

→

B

has a quasi-inverse, then so do

(

f

∘

–

)

:

(

C

→

A

)

→

(

C

→

B

)

(

–

∘

f

)

:

(

B

→

C

)

→

(

A

→

C

)

.

Proof.

If

g

is a quasi-inverse of

f

, then

(

g

∘

–

)

and

(

–

∘

g

)

are quasi-inverses of

(

f

∘

–

)

and

(

–

∘

f

)

respectively.
∎

Lemma 4.2.9

.

If

f

:

A

→

B

has a quasi-inverse, then the types

rinv

⁢

(

f

)

and

linv

⁢

(

f

)

are contractible.

Proof.

By function extensionality, we have

𝗅𝗂𝗇𝗏

(

f

)

≃

∑

g

:

B

→

A

(

g

∘

f

=

𝗂𝖽

A

)

.

But this is the fiber of

(

–

∘

f

)

over

𝗂𝖽

A

, and so
by Lemma 4.2.8 (

http://planetmath.org/42halfadjointequivalences#Thmprelem3

),

Theorem

4.2.3 (

http://planetmath.org/42halfadjointequivalences#Thmprethm1

),Theorem 4.2.6 (

http://planetmath.org/42halfadjointequivalences#Thmprethm2

), it is contractible.
Similarly,

𝗋𝗂𝗇𝗏

⁢

(

f

)

is equivalent to the fiber of

(

f

∘

–

)

over

𝗂𝖽

B

and hence contractible.
∎

Next we define the types which put together the other homotopy with the additional coherence datum.

Definition 4.2.10

.

For

f

:

A

→

B

, a left inverse

(

g

,

η

)

:

linv

⁢

(

f

)

, and a right inverse

(

g

,

ϵ

)

:

rinv

⁢

(

f

)

, we denote

𝗅𝖼𝗈𝗁

f

⁢

(

g

,

η

)

:

≡

∑

(

ϵ

:

f

∘

g

∼

𝗂𝖽

B

)

∏

(

y

:

B

)

g

(

ϵ

y

)

=

η

(

g

y

)

,

𝗋𝖼𝗈𝗁

f

⁢

(

g

,

ϵ

)

:

≡

∑

(

η

:

g

∘

f

∼

𝗂𝖽

A

)

∏

(

x

:

A

)

f

(

η

x

)

=

ϵ

(

f

x

)

.

Lemma 4.2.11

.

For any

f

,

g

,

ϵ

,

η

, we have

𝗅𝖼𝗈𝗁

f

⁢

(

g

,

η

)

≃

∏

y

:

B

(

f

⁢

g

⁢

y

,

η

⁢

(

g

⁢

y

)

)

=

𝖿𝗂𝖻

g

⁢

(

g

⁢

y

)

(

y

,

𝗋𝖾𝖿𝗅

g

⁢

y

)

,

𝗋𝖼𝗈𝗁

f

⁢

(

g

,

ϵ

)

≃

∏

x

:

A

(

g

⁢

f

⁢

x

,

ϵ

⁢

(

f

⁢

x

)

)

=

𝖿𝗂𝖻

f

⁢

(

f

⁢

x

)

(

x

,

𝗋𝖾𝖿𝗅

f

⁢

x

)

.

Proof.

Using Lemma 4.2.5 (

http://planetmath.org/42halfadjointequivalences#Thmprelem2

).
∎

Lemma 4.2.12

.

If

f

is a half adjoint equivalence, then for any

(

g

,

ϵ

)

:

rinv

⁢

(

f

)

, the type

rcoh

f

⁢

(

g

,

ϵ

)

is contractible.

Proof.

By Lemma 4.2.11 (

http://planetmath.org/42halfadjointequivalences#Thmprelem5

) and the fact that dependent function types

preserve

contractible spaces, it suffices to show that for each

x

:

A

, the type

(

g

⁢

f

⁢

x

,

ϵ

⁢

(

f

⁢

x

)

)

=

𝖿𝗂𝖻

f

⁢

(

f

⁢

x

)

(

x

,

𝗋𝖾𝖿𝗅

f

⁢

x

)

is contractible.
But by Theorem 4.2.6 (

http://planetmath.org/42halfadjointequivalences#Thmprethm2

),

𝖿𝗂𝖻

f

⁢

(

f

⁢

x

)

is contractible, and any

path space

of a contractible space is itself contractible.
∎

Theorem 4.2.13

.

For any

f

:

A

→

B

, the type

ishae

⁢

(

f

)

is a mere proposition.

Proof.

By Lemma 3.11.3 (

http://planetmath.org/311contractibility#Thmprelem1

) it suffices to assume

f

to be a half adjoint equivalence and show that

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

is contractible.
Now by

associativity

of

Σ

(

http://planetmath.org/node/87641

Exercise 2.10), the type

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

is equivalent to

∑

u

:

𝗋𝗂𝗇𝗏

⁢

(

f

)

𝗋𝖼𝗈𝗁

f

⁢

(

𝗉𝗋

1

⁢

(

u

)

,

𝗉𝗋

2

⁢

(

u

)

)

.

But by Lemma 4.2.9 (

http://planetmath.org/42halfadjointequivalences#Thmprelem4

),Lemma 4.2.12 (

http://planetmath.org/42halfadjointequivalences#Thmprelem6

) and the fact that

Σ

preserves

contractibility

, the latter type is also contractible.
∎

Thus, we have shown that

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

has all three desiderata for the type

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

.
In the next two

sections

we consider a couple of other possibilities.

Title

4.2 Half adjoint equivalences

\metatable

Generated on Thu Feb 8 19:35:34 2018 by

LaTeXML