4.6 Surjections and embeddings

4.6 Surjections and embeddings

When

A

and

B

are sets and

f

:

A

→

B

is an

equivalence

, we also call it as

isomorphism

or a

bijection

.
(We avoid these words for types that are not sets, since in

homotopy theory

and higher category theory they often denote a stricter notion of “sameness” than

homotopy equivalence

.)
In

set theory

, a

function

is a bijection just when it is both

injective

and

surjective

.
The same is true in

type theory

, if we formulate these conditions appropriately.
For clarity, when dealing with types that are not sets, we will speak of

embeddings

instead of injections.

Definition 4.6.1

.

Let

f

:

A

→

B

.

1.

We say

f

is

surjective

(or a

surjection

)
if for every

b

:

B

we have

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

.

2.

We say

f

is a

embedding

if for every

x

,

y

:

A

the function

𝖺𝗉

f

:

(

x

=

A

y

)

→

(

f

(

x

)

=

B

f

(

y

)

)

is an equivalence.

In other words,

f

is surjective if every fiber of

f

is merely inhabited, or equivalently if for all

b

:

B

there merely exists an

a

:

A

such that

f

⁢

(

a

)

=

b

.
In traditional logical notation,

f

is surjective if

∀

(

b

:

B

)

.

∃

(

a

:

A

)

.

(

f

(

a

)

=

b

)

.
This must be distinguished from the stronger assertion that

∏

(

b

:

B

)

∑

(

a

:

A

)

(

f

(

a

)

=

b

)

; if this holds we say that

f

is a

split surjection

.

If

A

and

B

are sets, then by Lemma 3.3.3 (

http://planetmath.org/33merepropositions#Thmprelem2

),

f

is an embedding just when

∏

x

,

y

:

A

(

f

(

x

)

=

B

f

(

y

)

)

→

(

x

=

A

y

)

.

(4.6.2)

In this case we say that

f

is

injective

,
or an

injection

.
We avoid these word for types that are not sets, because they might be interpreted as (

4.6.2

), which is an ill-behaved notion for non-sets.
It is also true that any function between sets is surjective if and only if it is an

epimorphism

in a suitable sense, but this also fails for more general types, and surjectivity is generally the more important notion.

Theorem 4.6.3

.

A function

f

:

A

→

B

is an equivalence if and only if it is both surjective and an embedding.

Proof.

If

f

is an equivalence, then each

𝖿𝗂𝖻

f

⁢

(

b

)

is

contractible

, hence so is

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

, so

f

is surjective.
And we showed in

Theorem

2.11.1 (

http://planetmath.org/211identitytype#Thmprethm1

) that any equivalence is an embedding.

Conversely

, suppose

f

is a surjective embedding.
Let

b

:

B

; we show that

∑

(

x

:

A

)

(

f

(

x

)

=

b

)

is contractible.
Since

f

is surjective, there merely exists an

a

:

A

such that

f

⁢

(

a

)

=

b

.
Thus, the fiber of

f

over

b

is inhabited; it remains to show it is a mere proposition.
For this, suppose given

x

,

y

:

A

with

p

:

f

⁢

(

x

)

=

b

and

q

:

f

⁢

(

y

)

=

b

.
Then since

𝖺𝗉

f

is an equivalence, there exists

r

:

x

=

y

with

𝖺𝗉

f

⁢

(

r

)

=

p

⁢

\centerdot

⁢

q

-

1

.
However, using the

characterization

of paths in

Σ

-types, the latter equality rearranges to

r

*

(

p

)

=

q

.
Thus, together with

r

it exhibits

(

x

,

p

)

=

(

y

,

q

)

in the fiber of

f

over

b

.
∎

Corollary 4.6.4

.

For any

f

:

A

→

B

we have

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

≃

(

𝗂𝗌𝖤𝗆𝖻𝖾𝖽𝖽𝗂𝗇𝗀

⁢

(

f

)

×

𝗂𝗌𝖲𝗎𝗋𝗃𝖾𝖼𝗍𝗂𝗏𝖾

⁢

(

f

)

)

.

Proof.

Being a surjection and an embedding are both mere propositions; now apply Lemma 3.3.3 (

http://planetmath.org/33merepropositions#Thmprelem2

).
∎

Of course, this cannot be used as a definition of “equivalence”, since the definition of embeddings refers to equivalences.
However, this characterization can still be useful; see §8.8 (

http://planetmath.org/88whiteheadstheoremandwhiteheadsprinciple

).
We will generalize it in

http://planetmath.org/node/87580

Chapter 7.

Title

4.6 Surjections and embeddings

\metatable

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