2.11 Identity type

2.11 Identity type

Just as the type

a

=

A

a

′

is characterized up to

isomorphism

, with
a separate “definition” for each

A

, there is no simple

characterization

of the type

p

=

a

=

A

a

′

q

of paths between
paths

p

,

q

:

a

=

A

a

′

.
However, our other general classes of

theorems

do extend to identity types, such as the fact that they respect

equivalence

.

Theorem 2.11.1

.

If

f

:

A

→

B

is an equivalence, then for all

a

,

a

′

:

A

, so is

𝖺𝗉

f

:

(

a

=

A

a

′

)

→

(

f

(

a

)

=

B

f

(

a

′

)

)

.

Proof.

Let

f

-

1

be a

quasi-inverse

of

f

, with

homotopies

α

:

∏

b

:

B

(

f

(

f

-

1

(

b

)

)

=

b

)

and

β

:

∏

a

:

A

(

f

-

1

(

f

(

a

)

)

=

a

)

.

The quasi-inverse of

𝖺𝗉

f

is, essentially,

𝖺𝗉

f

-

1

:

(

f

(

a

)

=

f

(

a

′

)

)

→

(

f

-

1

(

f

(

a

)

)

=

f

-

1

(

f

(

a

′

)

)

)

.

However, in order to obtain an element of

a

=

A

a

′

from

𝖺𝗉

f

-

1

⁢

(

q

)

, we must concatenate with the paths

β

a

-

1

and

β

a

′

on either side.
To show that this gives a quasi-inverse of

𝖺𝗉

f

, on one hand we must show that for any

p

:

a

=

A

a

′

we have

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

𝖺𝗉

f

⁢

(

p

)

)

⁢

\centerdot

⁢

β

a

′

=

p

.

This follows from the functoriality of

𝖺𝗉

and the naturality of homotopies, Lemma 2.2.4 (

http://planetmath.org/22functionsarefunctors#Thmprelem4

),Lemma 2.4.3 (

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

).
On the other hand, we must show that for any

q

:

f

⁢

(

a

)

=

B

f

⁢

(

a

′

)

we have

𝖺𝗉

f

⁢

(

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

q

)

⁢

\centerdot

⁢

β

a

′

)

=

q

.

The proof of this is a little more involved, but each step is again an application of Lemma 2.2.4 (

http://planetmath.org/22functionsarefunctors#Thmprelem4

),Lemma 2.4.3 (

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

) (or simply canceling

inverse

paths):

𝖺𝗉

f

⁢

(

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

q

)

⁢

\centerdot

⁢

β

a

′

)

=

α

f

⁢

(

a

)

-

1

⁢

\centerdot

⁢

α

f

⁢

(

a

)

⁢

\centerdot

⁢

𝖺𝗉

f

⁢

(

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

q

)

⁢

\centerdot

⁢

β

a

′

)

⁢

\centerdot

⁢

α

f

⁢

(

a

′

)

-

1

⁢

\centerdot

⁢

α

f

⁢

(

a

′

)

=

α

f

⁢

(

a

)

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

⁢

(

𝖺𝗉

f

-

1

⁢

(

𝖺𝗉

f

⁢

(

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

q

)

⁢

\centerdot

⁢

β

a

′

)

)

)

⁢

\centerdot

⁢

α

f

⁢

(

a

′

)

=

α

f

⁢

(

a

)

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

⁢

(

β

a

⁢

\centerdot

⁢

β

a

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

-

1

⁢

(

q

)

⁢

\centerdot

⁢

β

a

′

⁢

\centerdot

⁢

β

a

′

-

1

)

⁢

\centerdot

⁢

α

f

⁢

(

a

′

)

=

α

f

⁢

(

a

)

-

1

⁢

\centerdot

⁢

𝖺𝗉

f

⁢

(

𝖺𝗉

f

-

1

⁢

(

q

)

)

⁢

\centerdot

⁢

α

f

⁢

(

a

′

)

=

q

.

∎

Thus, if for some type

A

we have a full characterization of

a

=

A

a

′

, the type

p

=

a

=

A

a

′

q

is determined as well.
For example:

•

Paths

p

=

q

, where

p

,

q

:

w

=

A

×

B

w

′

, are

equivalent

to pairs of paths

𝖺𝗉

𝗉𝗋

1

⁢

p

=

𝗉𝗋

1

⁢

w

=

A

𝗉𝗋

1

⁢

w

′

𝖺𝗉

𝗉𝗋

1

⁢

q

and

𝖺𝗉

𝗉𝗋

2

⁢

p

=

𝗉𝗋

2

⁢

w

=

B

𝗉𝗋

2

⁢

w

′

𝖺𝗉

𝗉𝗋

2

⁢

q

.

•

Paths

p

=

q

, where

p

,

q

:

f

=

∏

(

x

:

A

)

B

⁢

(

x

)

g

, are equivalent to homotopies

∏

x

:

A

(

𝗁𝖺𝗉𝗉𝗅𝗒

(

p

)

(

x

)

=

f

⁢

(

x

)

=

g

⁢

(

x

)

𝗁𝖺𝗉𝗉𝗅𝗒

(

q

)

(

x

)

)

.

Next we consider transport in families of paths, i.e. transport in

C

:

A

→

𝒰

where each

C

⁢

(

x

)

is an identity type.
The simplest case is when

C

⁢

(

x

)

is a type of paths in

A

itself, perhaps with one

endpoint

fixed.

Lemma 2.11.2

.

For any

A

and

a

:

A

, with

p

:

x

1

=

x

2

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

(

a

=

x

)

⁢

(

p

,

q

)

=

q

⁢

\centerdot

⁢

p

for

q

:

a

=

x

1

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

(

x

=

a

)

⁢

(

p

,

q

)

=

p

-

1

⁢

\centerdot

⁢

q

for

q

:

x

1

=

a

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

(

x

=

x

)

⁢

(

p

,

q

)

=

p

-

1

⁢

\centerdot

⁢

q

⁢

\centerdot

⁢

p

for

q

:

x

1

=

x

1

.

Proof.

Path

induction

on

p

, followed by the unit laws for

composition

.
∎

In other words, transporting with

x

↦

c

=

x

is post-composition, and transporting with

x

↦

x

=

c

is contravariant pre-composition.
These may be familiar as the functorial actions of the covariant and contravariant hom-functors

hom

⁡

(

c

,

–

)

and

hom

⁡

(

–

,

c

)

in

category theory

.

Combining Lemma 2.11.2 (

http://planetmath.org/211identitytype#Thmprelem1

),Lemma 2.3.8 (

http://planetmath.org/23typefamiliesarefibrations#Thmprelem7

), we obtain a more general form:

Theorem 2.11.3

.

For

f

,

g

:

A

→

B

, with

p

:

a

=

A

a

′

and

q

:

f

⁢

(

a

)

=

B

g

⁢

(

a

)

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

f

⁢

(

x

)

=

B

g

⁢

(

x

)

⁢

(

p

,

q

)

=

f

⁢

(

a

′

)

=

g

⁢

(

a

′

)

(

𝖺𝗉

f

⁢

p

)

-

1

⁢

\centerdot

⁢

q

⁢

\centerdot

⁢

𝖺𝗉

g

⁢

p

.

Because

𝖺𝗉

(

x

↦

x

)

is the

identity function

and

𝖺𝗉

(

x

↦

c

)

(where

c

is a

constant

) is

𝗋𝖾𝖿𝗅

c

, Lemma 1 (

http://planetmath.org/211identitytype#Thmlem1

) is a special case.
A yet more general version is when

B

can be a family of types indexed on

A

:

Theorem 2.11.4

.

Let

B

:

A

→

U

and

f

,

g

:

∏

(

x

:

A

)

B

⁢

(

x

)

, with

p

:

a

=

A

a

′

and

q

:

f

⁢

(

a

)

=

B

⁢

(

a

)

g

⁢

(

a

)

.
Then we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

f

⁢

(

x

)

=

B

⁢

(

x

)

g

⁢

(

x

)

⁢

(

p

,

q

)

=

(

𝖺𝗉𝖽

f

⁢

(

p

)

)

-

1

⁢

\centerdot

⁢

𝖺𝗉

(

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

A

⁢

p

)

⁢

(

q

)

⁢

\centerdot

⁢

𝖺𝗉𝖽

g

⁢

(

p

)

.

Finally, as in §2.9 (

http://planetmath.org/29pitypesandthefunctionextensionalityaxiom

), for families of identity types there is another equivalent characterization of dependent paths.

Theorem 2.11.5

.

For

p

:

a

=

A

a

′

with

q

:

a

=

a

and

r

:

a

′

=

a

′

, we have

(

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

x

↦

(

x

=

x

)

(

p

,

q

)

=

r

)

≃

(

q

\centerdot

p

=

p

\centerdot

r

)

.

Proof.

Path induction on

p

, followed by the fact that composing with the unit equalities

q

⁢

\centerdot

⁢

1

=

q

and

r

=

1

⁢

\centerdot

⁢

r

is an equivalence.
∎

There are more general equivalences involving the application of functions, akin to Theorem 2.11.3 (

http://planetmath.org/211identitytype#S0.Thmprethm2

),Theorem 2.11.4 (

http://planetmath.org/211identitytype#S0.Thmprethm3

).

Title

2.11 Identity type

\metatable

Generated on Thu Feb 8 19:33:00 2018 by

LaTeXML