10.1.2 Images

10.1.2 Images

Next, we show that

𝒮

⁢

e

⁢

t

is a

regular category

, i.e.:

1.

𝒮

⁢

e

⁢

t

is

finitely complete

.

2.

The

kernel pair

𝗉𝗋

1

,

𝗉𝗋

2

:

(

∑

(

x

,

y

:

A

)

f

(

x

)

=

f

(

y

)

)

→

A

of any

function

f

:

A

→

B

has a

coequalizer

.

3.

Pullbacks

of

regular epimorphisms

are again regular epimorphisms.

Recall that a

regular epimorphism

is a

morphism

that is the coequalizer of

some

pair of maps.
Thus in

3

the pullback of a coequalizer is required to again be a coequalizer, but not necessarily of the pulled-back pair.

The obvious candidate for the coequalizer of the kernel pair of

f

:

A

→

B

is the

image

of

f

, as defined in

\autoref

sec:image-factorization.
Recall that we defined

𝗂𝗆

(

f

)

:

≡

∑

(

b

:

B

)

∥

𝖿𝗂𝖻

f

(

b

)

∥

, with functions

f

~

:

A

→

𝗂𝗆

⁢

(

f

)

and

i

f

:

𝗂𝗆

⁢

(

f

)

→

B

defined by

f

~

:

≡

λ

a

.

(

f

(

a

)

,

|

(

a

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

|

)

i

f

:

≡

𝗉𝗋

1

fitting into a

diagram

:

\xymatrix

*

*

[

l

]

∑

x

,

y

:

A

f

(

x

)

=

f

(

y

)

\ar

@

<

0.25

e

m

>

[

r

]

𝗉𝗋

1

\ar

@

<

-

0.25

e

m

>

[

r

]

𝗉𝗋

2

&

A

\ar

[

r

]

(

0.4

)

f

~

\ar

[

r

d

]

f

&

𝗂𝗆

(

f

)

\ar

@

.

.

>

[

d

]

i

f

&

&

B

Recall that a function

f

:

A

→

B

is called

surjective

if

∀

(

b

:

B

)

.

∥

𝖿𝗂𝖻

f

(

b

)

∥

,

or equivalently

∀

(

b

:

B

)

.

∃

(

a

:

A

)

.

f

(

a

)

=

b

.
We have also said that a function

f

:

A

→

B

between sets is called

injective

if

∀

(

a

,

a

′

:

A

)

.

(

f

(

a

)

=

f

(

a

′

)

)

⇒

(

a

=

a

′

)

, or equivalently if each of its fibers is a mere proposition.
Since these are the

(

-

1

)

-

connected

and

(

-

1

)

-truncated maps in the sense of

\autoref

cha:hlevels, the general theory there implies that

f

~

above is surjective and

i

f

is injective, and that this factorization is stable under pullback.

We now identify surjectivity and injectivity with the appropriate category-theoretic notions.
First we observe that

categorical

monomorphisms

and

epimorphisms

have a slightly

stronger

equivalent

formulation.

Lemma 10.1.1

.

For a morphism

f

:

hom

A

⁡

(

a

,

b

)

in a

category

A

, the following are equivalent.

1.

f

is a

monomorphism

:
for all

x

:

A

and

g

,

h

:

hom

A

⁡

(

x

,

a

)

, if

f

∘

g

=

f

∘

h

then

g

=

h

.

2.

(If

A

has pullbacks) the

diagonal map

a

→

a

×

b

a

is an

isomorphism

.

3.

For all

x

:

A

and

k

:

hom

A

⁡

(

x

,

b

)

, the type

∑

(

h

:

hom

A

(

x

,

a

)

)

(

k

=

f

∘

h

)

is a mere proposition.

4.

For all

x

:

A

and

g

:

hom

A

⁡

(

x

,

a

)

, the type

∑

(

h

:

hom

A

(

x

,

a

)

)

(

f

∘

g

=

f

∘

h

)

is

contractible

.

Proof.

The

equivalence

of conditions

1

and

2

is standard

category theory

.
Now consider the function

(

f

∘

–

)

:

hom

A

⁡

(

x

,

a

)

→

hom

A

⁡

(

x

,

b

)

between sets.
Condition

1

says that it is injective, while

3

says that its fibers are mere propositions; hence they are equivalent.
And

3

implies

4

by taking

k

:

≡

f

∘

g

and recalling that an inhabited mere proposition is contractible.
Finally,

4

implies

1

since if

p

:

f

∘

g

=

f

∘

h

, then

(

g

,

𝗋𝖾𝖿𝗅

)

and

(

h

,

p

)

both inhabit the type in

4

, hence are equal and so

g

=

h

.
∎

Lemma 10.1.2

.

A function

f

:

A

→

B

between sets is injective if and only if it is a monomorphism in

S

⁢

e

⁢

t

.

Proof.

Left to the reader.
∎

Of course, an

epimorphism

is a monomorphism in the opposite category.
We now show that in

𝒮

⁢

e

⁢

t

, the epimorphisms are precisely the surjections, and also precisely the coequalizers (regular epimorphisms).

The coequalizer of a pair of maps

f

,

g

:

A

→

B

in

𝒮

⁢

e

⁢

t

is defined as the 0-truncation of a general (

homotopy

) coequalizer.
For clarity, we may call this the

set-coequalizer

.
It is convenient to express its

universal property

as follows.

Lemma 10.1.3

.

Let

f

,

g

:

A

→

B

be functions between sets

A

and

B

. The
set-coequalizer

c

f

,

g

:

B

→

Q

has the property that, for any set

C

and any

h

:

B

→

C

with

h

∘

f

=

h

∘

g

, the type

∑

k

:

Q

→

C

(

k

∘

c

f

,

g

=

h

)

is contractible.

Lemma 10.1.4

.

For any function

f

:

A

→

B

between sets, the following are equivalent:

1.

f

is an epimorphism.

2.

Consider the pushout diagram

\xymatrix

⁢

A

⁢

\ar

⁢

[

r

]

f

⁢

\ar

⁢

[

d

]

⁢

&

⁢

B

⁢

\ar

⁢

[

d

]

ι

⁢

𝟏

⁢

\ar

⁢

[

r

]

t

⁢

&

⁢

C

f

in

𝒮

⁢

e

⁢

t

defining the mapping cone. Then the type

C

f

is contractible.

3.

f

is surjective.

Proof.

Let

f

:

A

→

B

be a function between sets, and suppose it to be an epimorphism; we show

C

f

is contractible.
The constructor

𝟏

→

C

f

of

C

f

gives us an

element

t

:

C

f

.
We have to show that

∏

x

:

C

f

x

=

t

.

Note that

x

=

t

is a mere proposition, hence we can use

induction

on

C

f

.
Of course when

x

is

t

we have

𝗋𝖾𝖿𝗅

t

:

t

=

t

, so it suffices to find

I

0

:

∏

b

:

B

ι

⁢

(

b

)

=

t

I

1

:

∏

a

:

A

α

1

⁢

(

a

)

-

1

⁢

\centerdot

⁢

I

0

⁢

(

f

⁢

(

a

)

)

=

𝗋𝖾𝖿𝗅

t

.

where

ι

:

B

→

C

f

and

α

1

:

∏

(

a

:

A

)

ι

⁢

(

f

⁢

(

a

)

)

=

t

are the other constructors
of

C

f

. Note that

α

1

is a homotopy from

ι

∘

f

to

𝖼𝗈𝗇𝗌𝗍

t

∘

f

, so we find the elements

(

ι

,

𝗋𝖾𝖿𝗅

ι

∘

f

)

,

(

𝖼𝗈𝗇𝗌𝗍

t

,

α

1

)

:

∑

h

:

B

→

C

f

ι

∘

f

∼

h

∘

f

.

By the dual of

\autoref

thm:mono

4

(and function extensionality), there is a path

γ

:

(

ι

,

𝗋𝖾𝖿𝗅

ι

∘

f

)

=

(

𝖼𝗈𝗇𝗌𝗍

t

,

α

1

)

.

Hence, we may define

I

0

(

b

)

:

≡

𝗁𝖺𝗉𝗉𝗅𝗒

(

𝖺𝗉

𝗉𝗋

1

(

γ

)

,

b

)

:

ι

(

b

)

=

t

.
We also have

𝖺𝗉

𝗉𝗋

2

(

γ

)

:

𝖺𝗉

𝗉𝗋

1

(

γ

)

*

(

𝗋𝖾𝖿𝗅

ι

∘

f

)

=

α

1

.

This transport involves precomposition with

f

, which commutes with

𝗁𝖺𝗉𝗉𝗅𝗒

.
Thus, from transport in path types we obtain

I

0

⁢

(

f

⁢

(

a

)

)

=

α

1

⁢

(

a

)

for any

a

:

A

, which gives us

I

1

.

Now suppose

C

f

is contractible; we show

f

is surjective.
We first

construct

a type family

P

:

C

f

→

𝖯𝗋𝗈𝗉

by

recursion

on

C

f

, which is valid since

𝖯𝗋𝗈𝗉

is a set.
On the point constructors, we define

P

⁢

(

t

)

:

≡

𝟏

P

⁢

(

ι

⁢

(

b

)

)

:

≡

∥

𝖿𝗂𝖻

f

(

b

)

∥

.

To

complete

the construction of

P

, it remains to give a path

∥

𝖿𝗂𝖻

f

(

f

(

a

)

)

∥

=

𝖯𝗋𝗈𝗉

𝟏

for all

a

:

A

.
However,

∥

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

∥

is inhabited by

(

f

⁢

(

a

)

,

𝗋𝖾𝖿𝗅

f

⁢

(

a

)

)

.
Since it is a mere proposition, this means it is contractible — and thus equivalent, hence equal, to

𝟏

.
This completes the definition of

P

.
Now, since

C

f

is assumed to be contractible, it follows that

P

⁢

(

x

)

is equivalent to

P

⁢

(

t

)

for any

x

:

C

f

.
In particular,

P

(

ι

(

b

)

)

≡

∥

𝖿𝗂𝖻

f

(

b

)

∥

is equivalent to

P

⁢

(

t

)

≡

𝟏

for each

b

:

B

, and hence contractible.
Thus,

f

is surjective.

Finally, suppose

f

:

A

→

B

to be surjective, and consider a set

C

and two functions

g

,

h

:

B

→

C

with the property that

g

∘

f

=

h

∘

f

. Since

f

is assumed to be surjective, for all

b

:

B

the type

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

is contractible.
Thus we have the following equivalences:

∏

b

:

B

(

g

(

b

)

=

h

(

b

)

)

≃

∏

b

:

B

(

∥

𝖿𝗂𝖻

f

(

b

)

∥

→

(

g

(

b

)

=

h

(

b

)

)

)

≃

∏

b

:

B

(

𝖿𝗂𝖻

f

(

b

)

→

(

g

(

b

)

=

h

(

b

)

)

)

≃

∏

(

b

:

B

)

∏

(

a

:

A

)

∏

(

p

:

f

(

a

)

=

b

)

g

⁢

(

b

)

=

h

⁢

(

b

)

≃

∏

a

:

A

g

⁢

(

f

⁢

(

a

)

)

=

h

⁢

(

f

⁢

(

a

)

)

using on the second line the fact that

g

⁢

(

b

)

=

h

⁢

(

b

)

is a mere proposition, since

C

is a set.
But by

assumption

, there is an element of the latter type.
∎

Theorem 10.1.5

.

The category

S

⁢

e

⁢

t

is

regular

. Moreover, surjective functions between sets are regular epimorphisms.

Proof.

It is a standard lemma in category theory that a category is regular as soon as it admits finite limits and a pullback-stable orthogonal
factorization system

(

ℰ

,

ℳ

)

with

ℳ

the monomorphisms, in which case

ℰ

consists automatically of
the regular epimorphisms.
(See e.g.

[

elephant

, A1.3.4]

.)
The

existence

of the

factorization system

was proved in

\autoref

thm:orth-fact.
∎

Lemma 10.1.6

.

Pullbacks of regular epis in

S

⁢

e

⁢

t

are regular epis.

Proof.

We showed in

\autoref

thm:stable-images that pullbacks of

n

-connected functions are

n

-connected.
By

\autoref

lem:images_are_coequalizers, it suffices to apply this when

n

=

-

1

.
∎

One of the

consequences

of

𝒮

⁢

e

⁢

t

being a regular category is that we have an “image”

operation

on subsets.
That is, given

f

:

A

→

B

, any subset

P

:

𝒫

⁢

(

A

)

(i.e. a

predicate

P

:

A

→

𝖯𝗋𝗈𝗉

) has an

image

which is a subset of

B

.
This can be defined directly as

\setof

y

:

B

|

∃

(

x

:

A

)

.

f

(

x

)

=

y

∧

P

(

x

)

, or indirectly as the image (in the previous sense) of the composite function

\setof

x

:

A

|

P

(

x

)

→

A

→

𝑓

B

.

We will also sometimes use the common notation

\setof

f

(

x

)

|

P

(

x

)

for the image of

P

.

Title

10.1.2 Images

\metatable

Generated on Thu Feb 8 19:29:08 2018 by

LaTeXML