10.1.3 Quotients

10.1.3 Quotients

Now that we know that

𝒮

⁢

e

⁢

t

is

regular

, to show that

𝒮

⁢

e

⁢

t

is exact, we need to show that every

equivalence relation

is effective.
In other words, given an equivalence
relation

R

:

A

→

A

→

𝖯𝗋𝗈𝗉

, there is a

coequalizer

c

R

of the pair

𝗉𝗋

1

,

𝗉𝗋

2

:

∑

(

x

,

y

:

A

)

R

⁢

(

x

,

y

)

→

A

and, moreover, the

𝗉𝗋

1

and

𝗉𝗋

2

form the

kernel pair

of

c

R

.

We have already seen, in

\autoref

sec:set-quotients, two general ways to

construct

the quotient of a set by an equivalence relation

R

:

A

→

A

→

𝖯𝗋𝗈𝗉

.
The first can be described as the set-coequalizer of the two projections

𝗉𝗋

1

,

𝗉𝗋

2

:

(

∑

x

,

y

:

A

R

⁢

(

x

,

y

)

)

→

A

.

The important property of such a quotient is the following.

Definition 10.1.1

.

A

relation

R

:

A

→

A

→

Prop

is said to be

effective

if the square

\xymatrix

∑

x

,

y

:

A

R

(

x

,

y

)

\ar

[

r

]

(

0.7

)

𝗉𝗋

1

\ar

[

d

]

𝗉𝗋

2

&

A

\ar

[

d

]

c

R

A

\ar

[

r

]

c

R

&

A

/

R

is a

pullback

.

Since the standard pullback of

c

R

and itself is

∑

(

x

,

y

:

A

)

(

c

R

(

x

)

=

c

R

(

y

)

)

, by

\autoref

thm:total-fiber-equiv this is

equivalent

to asking that the canonical transformation

∏

(

x

,

y

:

A

)

R

(

x

,

y

)

→

(

c

R

(

x

)

=

c

R

(

y

)

)

be a fiberwise

equivalence

.

Lemma 10.1.2

.

Suppose

(

A

,

R

)

is an equivalence relation. Then there is an equivalence

(

c

R

(

x

)

=

c

R

(

y

)

)

≃

R

(

x

,

y

)

for any

x

,

y

:

A

. In other words, equivalence relations are effective.

Proof.

We begin by extending

R

to a relation

R

~

:

A

/

R

→

A

/

R

→

𝖯𝗋𝗈𝗉

, which we will then show is equivalent
to the identity type on

A

/

R

. We define

R

~

by double

induction

on

A

/

R

(note that

𝖯𝗋𝗈𝗉

is a set by univalence for mere propositions). We
define

R

~

(

c

R

(

x

)

,

c

R

(

y

)

)

:

≡

R

(

x

,

y

)

. For

r

:

R

⁢

(

x

,

x

′

)

and

s

:

R

⁢

(

y

,

y

′

)

,
the

transitivity

and

symmetry

of

R

gives an equivalence from

R

⁢

(

x

,

y

)

to

R

⁢

(

x

′

,

y

′

)

. This

completes

the
definition of

R

~

.

It remains to show that

R

~

(

w

,

w

′

)

≃

(

w

=

w

′

)

for every

w

,

w

′

:

A

/

R

.
The direction

(

w

=

w

′

)

→

R

~

(

w

,

w

′

)

follows by transport once we show that

R

~

is

reflexive

, which is an easy induction.
The other direction

R

~

(

w

,

w

′

)

→

(

w

=

w

′

)

is a mere proposition, so since

c

R

:

A

→

A

/

R

is

surjective

, it suffices to assume that

w

and

w

′

are of the form

c

R

⁢

(

x

)

and

c

R

⁢

(

y

)

.
But in this case, we have the

canonical map

R

~

(

c

R

(

x

)

,

c

R

(

y

)

)

:

≡

R

(

x

,

y

)

→

(

c

R

(

x

)

=

c

R

(

y

)

)

.
(Note again the appearance of the encode-decode method.)
∎

The second construction of quotients is as the set of

equivalence classes

of

R

(a subset
of its

power set

):

A

\sslash

R

:

≡

\setof

P

:

A

→

𝖯𝗋𝗈𝗉

|

P

is an equivalence class of

R

This requires propositional resizing in order to remain in the same

universe

as

A

and

R

.

Note that if we regard

R

as a

function

from

A

to

A

→

𝖯𝗋𝗈𝗉

, then

A

⁢

\sslash

⁢

R

is equivalent to

𝗂𝗆

⁢

(

R

)

, as constructed in

\autoref

sec:image.
Now in

\autoref

lem:images_are_coequalizers we have shown that images are
coequalizers. In particular, we immediately get the coequalizer

diagram

\xymatrix

*

*

[

l

]

∑

x

,

y

:

A

R

(

x

)

=

R

(

y

)

\ar

@

<

0.25

e

m

>

[

r

]

𝗉𝗋

1

\ar

@

<

-

0.25

e

m

>

[

r

]

𝗉𝗋

2

&

A

\ar

[

r

]

&

A

\sslash

R

.

We can use this to give an alternative proof that any equivalence relation is effective and that the two definitions of quotients agree.

Theorem 10.1.3

.

For any function

f

:

A

→

B

between any two sets,
the relation

ker

⁡

(

f

)

:

A

→

A

→

Prop

given by

ker

(

f

,

x

,

y

)

:

≡

(

f

(

x

)

=

f

(

y

)

)

is effective.

Proof.

We will use that

𝗂𝗆

⁢

(

f

)

is the coequalizer of

𝗉𝗋

1

,

𝗉𝗋

2

:

(

∑

(

x

,

y

:

A

)

f

(

x

)

=

f

(

y

)

)

→

A

.
Note that the kernel pair of the function

c

f

:

≡

λ

a

.

(

f

(

a

)

,

∥

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

∥

)

:

A

→

𝗂𝗆

(

f

)

consists of the two projections

𝗉𝗋

1

,

𝗉𝗋

2

:

(

∑

x

,

y

:

A

c

f

(

x

)

=

c

f

(

y

)

)

→

A

.

For any

x

,

y

:

A

, we have equivalences

(

c

f

⁢

(

x

)

=

c

f

⁢

(

y

)

)

≃

(

∑

p

:

f

⁢

(

x

)

=

f

⁢

(

y

)

p

*

(

∥

(

x

,

𝗋𝖾𝖿𝗅

f

⁢

(

x

)

)

∥

)

=

∥

(

y

,

𝗋𝖾𝖿𝗅

f

⁢

(

x

)

)

∥

)

≃

(

f

(

x

)

=

f

(

y

)

)

,

where the last equivalence holds because

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

is a mere proposition for any

b

:

B

.
Therefore, we get that

(

∑

x

,

y

:

A

c

f

(

x

)

=

c

f

(

y

)

)

≃

(

∑

x

,

y

:

A

f

(

x

)

=

f

(

y

)

)

and hence we may conclude that

ker

⁡

f

is an effective relation
for any function

f

.
∎

Theorem 10.1.4

.

Equivalence relations are effective and there is an equivalence

A

/

R

≃

A

⁢

\sslash

⁢

R

.

Proof.

We need to analyze the coequalizer diagram

\xymatrix

*

*

[

l

]

∑

x

,

y

:

A

R

(

x

)

=

R

(

y

)

\ar

@

<

0.25

e

m

>

[

r

]

𝗉𝗋

1

\ar

@

<

-

0.25

e

m

>

[

r

]

𝗉𝗋

2

&

A

\ar

[

r

]

&

A

\sslash

R

By the univalence axiom, the type

R

⁢

(

x

)

=

R

⁢

(

y

)

is equivalent to the type of

homotopies

from

R

⁢

(

x

)

to

R

⁢

(

y

)

, which is
equivalent to

∏

(

z

:

A

)

R

⁢

(

x

,

z

)

≃

R

⁢

(

y

,

z

)

.

Since

R

is an equivalence relation, the latter space is equivalent to

R

⁢

(

x

,

y

)

. To
summarize, we get that

(

R

(

x

)

=

R

(

y

)

)

≃

R

(

x

,

y

)

, so

R

is effective since it is equivalent to an effective relation. Also,
the diagram

\xymatrix

*

*

[

l

]

∑

x

,

y

:

A

R

(

x

,

y

)

\ar

@

<

0.25

e

m

>

[

r

]

𝗉𝗋

1

\ar

@

<

-

0.25

e

m

>

[

r

]

𝗉𝗋

2

&

A

\ar

[

r

]

&

A

\sslash

R

.

is a coequalizer diagram. Since coequalizers are unique up to equivalence, it follows that

A

/

R

≃

A

⁢

\sslash

⁢

R

.
∎

We finish this

section

by mentioning a possible third construction of the quotient of a set

A

by an equivalence relation

R

.
Consider the

precategory

with objects

A

and hom-sets

R

; the type of objects of the Rezk completion
(see

\autoref

sec:rezk) of this precategory will then be the
quotient. The reader is invited to check the details.

Title

10.1.3 Quotients

\metatable

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