9.8 The structure identity principle

9.8 The structure identity principle

The

structure

identity

principle

is an informal principle
that expresses that

isomorphic

structures are identical. We aim to
prove a general abstract result which can be applied to a wide family
of notions of structure, where structures may be many-sorted or even
dependently-sorted, infinitary, or even higher order.

The simplest kind of single-sorted structure consists of a type with
no additional structure. The univalence axiom expresses the structure identity principle for that
notion of structure in a strong form: for types

A

,

B

, the
canonical

function

(

A

=

B

)

→

(

A

≃

B

)

is an equivalence.

We start with a

precategory

X

. In our application to
single-sorted

first order

structures,

X

will be the

category

of

𝒰

-small sets, where

𝒰

is a

univalent

type

universe

.

Definition 9.8.1

.

A

notion of structure

(

P

,

H

)

over

X

consists of the following.

1.

A type family

P

:

X

0

→

𝒰

.
For each

x

:

X

0

the elements of

P

⁢

x

are called

(

P

,

H

)

-structures

on

x

.

2.

For

x

,

y

:

X

0

and

α

:

P

⁢

x

,

β

:

P

⁢

y

, to each

f

:

hom

X

⁡

(

x

,

y

)

a mere proposition

H

α

⁢

β

⁢

(

f

)

.

If

H

α

⁢

β

⁢

(

f

)

is true, we say that

f

is a

(

P

,

H

)

-

homomorphism

from

α

to

β

.

3.

For

x

:

X

0

and

α

:

P

⁢

x

, we have

H

α

⁢

α

⁢

(

1

x

)

.

4.

For

x

,

y

,

z

:

X

0

and

α

:

P

⁢

x

,

β

:

P

⁢

y

,

γ

:

P

⁢

z

,
if

f

:

hom

X

⁡

(

x

,

y

)

, we have

H

α

⁢

β

⁢

(

f

)

→

H

β

⁢

γ

⁢

(

g

)

→

H

α

⁢

γ

⁢

(

g

∘

f

)

.

When

(

P

,

H

)

is a notion of structure, for

α

,

β

:

P

⁢

x

we define

(

α

≤

x

β

)

:

≡

H

α

⁢

β

(

1

x

)

.

By

3

and

4

, this is a preorder (

\autoref

ct:orders) with

P

⁢

x

its type of objects.
We say that

(

P

,

H

)

is a

standard notion of structure

if this preorder is in fact a partial order, for all

x

:

X

.

Note that for a standard notion of structure, each type

P

⁢

x

must actually be a set.
We now define, for any notion of structure

(

P

,

H

)

, a

precategory of

(

P

,

H

)

-structures

,

A

=

𝖲𝗍𝗋

(

P

,

H

)

⁢

(

X

)

.

•

The type of objects of

A

is the type

A

0

:

≡

∑

(

x

:

X

)

P

x

.
If

a

≡

(

x

,

α

)

:

A

0

, we may write

|

a

|

:

≡

x

.

•

For

(

x

,

α

)

:

A

0

and

(

y

,

β

)

:

A

0

, we define

hom

A

(

(

x

,

α

)

,

(

y

,

β

)

)

:

≡

\setof

f

:

x

→

y

|

H

α

⁢

β

(

f

)

.

The composition and identities are inherited from

X

; conditions

3

and

4

ensure that these lift to

A

.

Theorem 9.8.2

(Structure identity principle).

If

X

is a category and

(

P

,

H

)

is a standard notion of structure over

X

, then the precategory

Str

(

P

,

H

)

⁢

(

X

)

is a category.

Proof.

By the definition of equality in dependent pair types, to give an equality

(

x

,

α

)

=

(

y

,

β

)

consists of

•

An equality

p

:

x

=

y

, and

•

An equality

p

*

(

α

)

=

β

.

Since

P

is set-valued, the latter is a mere proposition.
On the other hand, it is easy to see that an

isomorphism

(

x

,

α

)

≅

(

y

,

β

)

in

𝖲𝗍𝗋

(

P

,

H

)

⁢

(

X

)

consists of

•

An isomorphism

f

:

x

≅

y

in

X

, such that

•

H

α

⁢

β

⁢

(

f

)

and

H

β

⁢

α

⁢

(

f

-

1

)

.

Of course, the second of these is also a mere proposition.
And since

X

is a category, the function

(

x

=

y

)

→

(

x

≅

y

)

is an equivalence.
Thus, it will suffice to show that for any

p

:

x

=

y

and for any

(

α

:

P

⁢

x

)

,

(

β

:

P

⁢

y

)

, we have

p

*

(

α

)

=

β

if and only if both

H

α

⁢

β

⁢

(

𝗂𝖽𝗍𝗈𝗂𝗌𝗈

⁢

(

p

)

)

and

H

β

⁢

α

⁢

(

𝗂𝖽𝗍𝗈𝗂𝗌𝗈

⁢

(

p

)

-

1

)

.

The “only if”

direction

is just the

existence

of the function

𝗂𝖽𝗍𝗈𝗂𝗌𝗈

for the category

𝖲𝗍𝗋

(

P

,

H

)

⁢

(

X

)

.
For the “if” direction, by

induction

on

p

we may assume that

y

≡

x

and

p

≡

𝗋𝖾𝖿𝗅

x

.
However, in this case

𝗂𝖽𝗍𝗈𝗂𝗌𝗈

⁢

(

p

)

≡

1

x

and therefore

𝗂𝖽𝗍𝗈𝗂𝗌𝗈

⁢

(

p

)

-

1

=

1

x

.
Thus,

α

≤

x

β

and

β

≤

x

α

, which implies

α

=

β

since

(

P

,

H

)

is a standard notion of structure.
∎

As an example, this methodology gives an alternative way to express the proof of

\autoref

ct:functor-cat.

Example 9.8.3

.

Let

A

be a precategory and

B

a category.
There is a precategory

B

A

0

whose objects are functions

A

0

→

B

0

, and whose set of

morphisms

from

F

0

:

A

0

→

B

0

to

G

0

:

A

0

→

B

0

is

∏

(

a

:

A

0

)

hom

B

⁡

(

F

0

⁢

a

,

G

0

⁢

a

)

.
Composition and identities are inherited directly from those in

B

.
It is easy to show that

γ

:

hom

B

A

0

⁡

(

F

0

,

G

0

)

is an isomorphism exactly when each

component

γ

a

is an isomorphism, so that we have

(

F

0

≅

G

0

)

≃

∏

(

a

:

A

0

)

(

F

0

a

≅

G

0

a

)

.
Moreover, the map

idtoiso

:

(

F

0

=

G

0

)

→

(

F

0

≅

G

0

)

of

B

A

0

is equal to the composite

(

F

0

=

G

0

)

⟶

∏

a

:

A

0

(

F

0

a

=

G

0

a

)

⟶

∏

a

:

A

0

(

F

0

a

≅

G

0

a

)

⟶

(

F

0

≅

G

0

)

in which the first map is an equivalence by function extensionality, the second because it is a dependent product of equivalences (since

B

is a category), and the third as remarked above.
Thus,

B

A

0

is a category.

Now we define a notion of structure on

B

A

0

for which

P

⁢

(

F

0

)

is the type of

operations

F

:

∏

(

a

,

a

′

:

A

0

)

hom

A

⁡

(

a

,

a

′

)

→

hom

B

⁡

(

F

0

⁢

a

,

F

0

⁢

a

′

)

which extend

F

0

to a

functor

(i.e. preserve composition and identities).
This is a set since each

hom

B

⁡

(

–

,

–

)

is so.
Given such

F

and

G

, we define

γ

:

hom

B

A

0

⁡

(

F

0

,

G

0

)

to be a homomorphism if it forms a

natural transformation

.
In

\autoref

ct:functor-precat we essentially verified that this is a notion of structure.
Moreover, if

F

and

F

′

are both structures on

F

0

and the identity is a natural transformation from

F

to

F

′

, then for any

f

:

hom

A

⁡

(

a

,

a

′

)

we have

F

′

⁢

f

=

F

′

⁢

f

∘

1

F

0

⁢

a

=

1

F

0

⁢

a

∘

F

⁢

f

=

F

⁢

f

.
Applying function extensionality, we conclude

F

=

F

′

.
Thus, we have a

standard

notion of structure, and so by

\autoref

thm:sip, the precategory

B

A

is a category.

As another example, we consider categories of structures for a first-order

signature

.
We define a

first-order signature

,

Ω

, to consist of sets

Ω

0

and

Ω

1

of function symbols,

ω

:

Ω

0

, and relation symbols,

ω

:

Ω

1

, each having an arity

|

ω

|

that is a set.
An

Ω

-structure

a

consists of a set

|

a

|

together with an assignment of an

|

ω

|

-ary function

ω

a

:

|

a

|

|

ω

|

→

|

a

|

on

|

a

|

to each function symbol,

ω

, and an assignment of an

|

ω

|

-ary

relation

ω

a

on

|

a

|

, assigning a mere proposition

ω

a

⁢

x

to each

x

:

|

a

|

|

ω

|

, to each relation symbol.
And given

Ω

-structures

a

,

b

, a function

f

:

|

a

|

→

|

b

|

is a

homomorphism

a

→

b

if it preserves the structure; i.e. if for each symbol

ω

of the signature and each

x

:

|

a

|

|

ω

|

,

1.

f

⁢

(

ω

a

⁢

x

)

=

ω

b

⁢

(

f

∘

x

)

if

ω

:

Ω

0

, and

2.

ω

a

⁢

x

→

ω

b

⁢

(

f

∘

x

)

if

ω

:

Ω

1

.

Note that each

x

:

|

a

|

|

ω

|

is a function

x

:

|

ω

|

→

|

a

|

so that

f

∘

x

:

b

ω

.

Now we assume given a (univalent) universe

𝒰

and a

𝒰

-small signature

Ω

; i.e.

|

Ω

|

is a

𝒰

-small set and, for each

ω

:

|

Ω

|

, the set

|

ω

|

is

𝒰

-small.
Then we have the category

𝒮

⁢

e

⁢

t

𝒰

of

𝒰

-small sets. We want to define the precategory of

𝒰

-small

Ω

-structures over

𝒮

⁢

e

⁢

t

𝒰

and use

\autoref

thm:sip to show that it is a category.

We use the first order signature

Ω

to give us a standard notion of structure

(

P

,

H

)

over

𝒮

⁢

e

⁢

t

𝒰

.

Definition 9.8.4

.

1.

For each

𝒰

-small set

x

define

P

x

:

≡

P

0

x

×

P

1

x

.

Here

P

0

⁢

x

:

≡

∏

ω

:

Ω

0

x

|

ω

|

→

x

,

and

P

1

⁢

x

:

≡

∏

ω

:

Ω

1

x

|

ω

|

→

𝖯𝗋𝗈𝗉

𝒰

,

2.

For

𝒰

-small sets

x

,

y

and

α

:

P

ω

⁢

x

,

β

:

P

ω

⁢

y

,

f

:

x

→

y

, define

H

α

⁢

β

(

f

)

:

≡

H

0

,

α

⁢

β

(

f

)

∧

H

1

,

α

⁢

β

(

f

)

.

Here

H

0

,

α

⁢

β

⁢

(

f

)

:

≡

∀

(

ω

:

Ω

0

)

.

∀

(

u

:

x

|

ω

|

)

.

f

(

α

u

)

=

β

(

f

∘

u

)

,

and

H

1

,

α

⁢

β

⁢

(

f

)

:

≡

∀

(

ω

:

Ω

1

)

.

∀

(

u

:

x

|

ω

|

)

.

α

u

→

β

(

f

∘

u

)

.

It is now routine to check that

(

P

,

H

)

is a standard notion of structure over

𝒮

⁢

e

⁢

t

𝒰

and hence we may use

\autoref

thm:sip to get that the precategory

S

⁢

t

⁢

r

(

P

,

H

)

⁢

(

𝒮

⁢

e

⁢

t

𝒰

)

is a category. It only remains to observe that this is essentially the same as the precategory of

𝒰

-small

Ω

-structures over

𝒮

⁢

e

⁢

t

𝒰

.

Title

9.8 The structure identity principle

\metatable

Generated on Thu Feb 8 19:43:00 2018 by

LaTeXML