8.8 Whiteheadâs theorem and Whiteheadâs principle

8.8 Whiteheadâs theorem and Whiteheadâs principle

In classical

homotopy theory

, a map

f

:

A

→

B

which induces an

isomorphism

π

n

⁢

(

A

,

a

)

≅

π

n

⁢

(

B

,

f

⁢

(

a

)

)

for all points

a

in

A

(and also an isomorphism

π

0

⁢

(

A

)

≅

π

0

⁢

(

B

)

) is necessarily a

homotopy equivalence

, as long as the spaces

A

and

B

are well-behaved (e.g. have the

homotopy types

of

CW-complexes

).
This is known as

Whitehead’s

theorem

.
In fact, the “ill-behaved” spaces for which Whitehead’s theorem fails are invisible to

type theory

.
Roughly, the well-behaved

topological spaces

suffice to present

∞

-

groupoids

,
and

homotopy type theory

deals with

∞

-groupoids directly rather than actual topological spaces.
Thus, one might expect that Whitehead’s theorem would be true in

univalent foundations

.

However, this is

not

the case: Whitehead’s theorem is not provable.
In fact, there are known models of type theory in which it fails to be true, although for entirely different reasons than its failure for ill-behaved topological spaces.
These models are “non-hypercomplete

∞

-toposes”
(see

[

lurie:higher-topoi

]

); roughly speaking, they consist of sheaves of

∞

-groupoids over

∞

-dimensional

base spaces

.

From a foundational point of view, therefore, we may speak of

Whitehead’s principle

as a “classicality axiom”, akin to

𝖫𝖤𝖬

and

𝖠𝖢

.
It may consistently be assumed, but it is not part of the computationally motivated type theory, nor does it hold in all natural models.
But when working from set-theoretic

foundations

, this principle is invisible: it cannot fail to be true in a world where

∞

-groupoids are built up out of sets (using topological spaces,

simplicial sets

, or any other such model).

This may seem odd, but actually it should not be surprising.
Homotopy type theory is the

abstract

theory of homotopy types, whereas the homotopy theory of topological spaces or simplicial sets in

set theory

is a

concrete

model of this theory, in the same way that the integers are a concrete model of the abstract theory of rings.
It is to be expected that any concrete model will have special properties which are not intrinsic to the corresponding abstract theory, but which we might sometimes want to assume as additional axioms (e.g. the integers are a

Principal Ideal Domain

, but not all rings are).

It is beyond the scope of this book to describe any models of type theory, so we will not explain how Whitehead’s principle might fail in some of them.
However, we can prove that it holds whenever the types involved are

n

-truncated for some finite

n

, by “downward”

induction

on

n

.
In

addition

to being of interest in its own right (for

instance

, it implies the essential uniqueness of Eilenberg–Mac Lane spaces), the proof of this result will hopefully provide some intuitive explanation for why we cannot hope to prove an analogous theorem without truncation hypotheses.

We begin with the following modification of

\autoref

thm:mono-surj-equiv, which will eventually supply the induction step in the proof of the truncated Whitehead’s principle.
It may be regarded as a type-theoretic,

∞

-groupoidal version of the classical statement that a fully

faithful

and essentially surjective functor is an

equivalence of categories

.

Theorem 8.8.1

.

Suppose

f

:

A

→

B

is a

function

such that

1.

∥

f

∥

0

:

∥

A

∥

0

→

∥

B

∥

0

is

surjective

, and

2.

for any

x

,

y

:

A

, the function

𝖺𝗉

f

:

(

x

=

A

y

)

→

(

f

(

x

)

=

B

f

(

y

)

)

is an

equivalence

.

Then

f

is an equivalence.

Proof.

Note that

2

is precisely the statement that

f

is an

embedding

, c.f.

\autoref

sec:mono-surj.
Thus, by

\autoref

thm:mono-surj-equiv, it suffices to show that

f

is surjective, i.e. that for any

b

:

B

we have

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

-

1

.
Suppose given

b

; then since

∥

f

∥

0

is surjective, there merely exists an

a

:

A

such that

∥

f

∥

0

(

|

a

|

0

)

=

|

b

|

0

.
And since our goal is a mere proposition, we may assume given such an

a

.
Then we have

|

f

(

a

)

|

0

=

∥

f

∥

0

(

|

a

|

0

)

=

|

b

|

0

, hence

∥

f

(

a

)

=

b

∥

-

1

.
Again, since our goal is still a mere proposition, we may assume

f

⁢

(

a

)

=

b

.
Hence

𝖿𝗂𝖻

f

⁢

(

b

)

is inhabited, and thus merely inhabited.
∎

Since

homotopy groups

are truncations of

loop spaces

, rather than

path spaces

, we need to modify this theorem to speak about these instead.

Corollary 8.8.2

.

Suppose

f

:

A

→

B

is a function such that

1.

∥

f

∥

0

:

∥

A

∥

0

→

∥

B

∥

0

is a

bijection

, and

2.

for any

x

:

A

, the function

𝖺𝗉

f

:

Ω

⁢

(

A

,

x

)

→

Ω

⁢

(

B

,

f

⁢

(

x

)

)

is an equivalence.

Then

f

is an equivalence.

Proof.

By

\autoref

thm:whitehead0, it suffices to show that

𝖺𝗉

f

:

(

x

=

A

y

)

→

(

f

(

x

)

=

B

f

(

y

)

)

is an equivalence for any

x

,

y

:

A

.
And by

\autoref

thm:equiv-inhabcod, we may assume

f

⁢

(

x

)

=

B

f

⁢

(

y

)

.
In particular,

|

f

(

x

)

|

0

=

|

f

(

y

)

|

0

, so since

∥

f

∥

0

is an equivalence, we have

|

x

|

0

=

|

y

|

0

, hence

|

x

=

y

|

-

1

.
Since we are trying to prove a mere proposition (

f

being an equivalence), we may assume given

p

:

x

=

y

.
But now the following square commutes up to

homotopy

:

\xymatrix

@C=3pc

⁢

Ω

⁢

(A,x)

\ar

[r]^-

⁢

–

⁢

\centerdot

⁢

p

\ar

[d]_

𝖺𝗉

f

&
(

x

=

A

y

)

\ar

[d]^

𝖺𝗉

f

⁢

Ω

⁢

(B,f(x))

\ar

[r]_-

⁢

–

⁢

\centerdot

⁢

f(p) &
(

f

⁢

(

x

)

=

B

f

⁢

(

y

)

).

The top and bottom maps are equivalences, and the left-hand map is so by

assumption

.
Hence, by the 2-out-of-3 property, so is the right-hand map.
∎

Now we can prove the truncated Whitehead’s principle.

Theorem 8.8.3

.

Suppose

A

and

B

are

n

-types and

f

:

A

→

B

is such that

1.

∥

f

∥

0

:

∥

A

∥

0

→

∥

B

∥

0

is an isomorphism, and

2.

π

k

⁢

(

f

)

:

π

k

⁢

(

A

,

x

)

→

π

k

⁢

(

B

,

f

⁢

(

x

)

)

is a bijection for all

k

≥

1

and all

x

:

A

.

Then

f

is an equivalence.

Condition

1

is almost the case of

2

when

k

=

0

, except that it makes no reference to any

basepoint

x

:

A

.

Proof.

We proceed by induction on

n

.
When

n

=

-

2

, the statement is

trivial

.
Thus, suppose it to be true for all functions between

n

-types, and let

A

and

B

be

(

n

+

1

)

-types and

f

:

A

→

B

as above.
The first condition in

\autoref

thm:whitehead1 holds by assumption, so it will suffice to show that for any

x

:

A

, the function

𝖺𝗉

f

:

Ω

⁢

(

A

,

x

)

→

Ω

⁢

(

B

,

f

⁢

(

x

)

)

is an equivalence.
However,

Ω

⁢

(

A

,

x

)

and

Ω

⁢

(

B

,

f

⁢

(

x

)

)

are

n

-types, and

π

k

⁢

(

𝖺𝗉

f

)

=

π

k

+

1

⁢

(

f

)

, so this follows from the

inductive hypothesis

.
∎

Note that if

A

and

B

are not

n

-types for any finite

n

, then there is no way for the induction to get started.

Corollary 8.8.4

.

If

A

is a

0

-

connected

n

-type and

π

k

⁢

(

A

,

a

)

=

0

for all

k

and

a

:

A

, then

A

is

contractible

.

Proof.

Apply

\autoref

thm:whiteheadn to the map

A

→

𝟏

.
∎

As an application, we can deduce the

converse

of

\autoref

thm:conn-pik.

Corollary 8.8.5

.

For

n

≥

0

, a map

f

:

A

→

B

is

n

-connected if and only if the following all hold:

1.

∥

f

∥

0

:

∥

A

∥

0

→

∥

B

∥

0

is an isomorphism.

2.

For any

a

:

A

and

k

≤

n

, the map

π

k

⁢

(

f

)

:

π

k

⁢

(

A

,

a

)

→

π

k

⁢

(

B

,

f

⁢

(

a

)

)

is an isomorphism.

3.

For any

a

:

A

, the map

π

n

+

1

⁢

(

f

)

:

π

n

+

1

⁢

(

A

,

a

)

→

π

n

+

1

⁢

(

B

,

f

⁢

(

a

)

)

is surjective.

Proof.

The “only if” direction is

\autoref

thm:conn-pik.
Conversely, by the

long exact sequence

of a

fibration

(

\autoref

thm:les),
the hypotheses imply that

π

k

⁢

(

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

)

=

0

for all

k

≤

n

and

a

:

A

, and that

∥

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

∥

0

is contractible.
Since

π

k

(

𝖿𝗂𝖻

f

(

f

(

a

)

)

)

=

π

k

(

∥

𝖿𝗂𝖻

f

(

f

(

a

)

)

∥

n

)

for

k

≤

n

, and

∥

𝖿𝗂𝖻

f

⁢

(

f

⁢

(

a

)

)

∥

n

is

n

-connected, by

\autoref

thm:whitehead-contr it is contractible for any

a

.

It remains to show that

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

n

is contractible for

b

:

B

not necessarily of the form

f

⁢

(

a

)

.
However, by assumption, there is

x

:

∥

A

∥

0

with

|

b

|

0

=

∥

f

∥

0

(

x

)

.
Since

contractibility

is a mere proposition, we may assume

x

is of the form

|

a

|

0

for

a

:

A

, in which case

|

b

|

0

=

∥

f

∥

0

(

|

a

|

0

)

=

|

f

(

a

)

|

0

, and therefore

∥

b

=

f

(

a

)

∥

-

1

.
Again since contractibility is a mere proposition, we may assume

b

=

f

⁢

(

a

)

, and the result follows.
∎

A map

f

such that

∥

f

∥

0

is a bijection and

π

k

⁢

(

f

)

is a bijection for all

k

is called

∞

-connected

or a

weak equivalence

.
This is

equivalent

to asking that

f

be

n

-connected for all

n

.
A type

Z

is called

∞

-truncated

or

hypercomplete

if the

induced map

(

–

∘

f

)

:

(

B

→

Z

)

→

(

A

→

Z

)

is an equivalence whenever

f

is

∞

-connected — that is, if

Z

thinks every

∞

-connected map is an equivalence.
Then if we want to assume Whitehead’s principle as an axiom, we may use either of the following equivalent forms.

•

Every

∞

-connected function is an equivalence.

•

Every type is

∞

-truncated.

In higher topos models,
the

∞

-truncated types form a reflective subuniverse in the sense of

\autoref

sec:modalities (the “hypercompletion” of an

(

∞

,

1

)

-topos), but we do not know whether this is
true in general.

It may not be obvious that there

are

any types which are not

n

-types for any

n

, but in fact there are.
Indeed, in classical homotopy theory,

𝕊

n

has this property for any

n

≥

2

.
We have not proven this fact in homotopy type theory yet, but there are other types which we can prove to have “

infinite

truncation level”.

Example 8.8.6

.

Suppose we have

B

:

N

→

U

such that for each

n

, the type

B

⁢

(

n

)

contains an

n

-loop which is not equal to

n

-fold

reflexivity

, say

p

n

:

Ω

n

⁢

(

B

⁢

(

n

)

,

b

n

)

with

p

n

≠

refl

b

n

n

.
(For instance, we could define

B

(

n

)

:

≡

S

n

, with

p

n

the image of

1

:

Z

under the isomorphism

π

n

⁢

(

S

n

)

≅

Z

.)
Consider

C

:

≡

∏

(

n

:

N

)

B

(

n

)

, with the point

c

:

C

defined by

c

(

n

)

:

≡

b

n

.
Since loop spaces commute with

products

, for any

m

we have

Ω

m

⁢

(

C

,

c

)

≃

∏

n

:

ℕ

Ω

m

⁢

(

B

⁢

(

n

)

,

b

n

)

.

Under this equivalence,

refl

c

m

corresponds to the function

(

n

↦

refl

b

n

m

)

.
Now define

q

m

in the right-hand type by

q

m

(

n

)

:

≡

{

p

n

m

=

n

𝗋𝖾𝖿𝗅

b

n

m

m

≠

n

.

If we had

q

m

=

(

n

↦

refl

b

n

m

)

, then we would have

p

n

=

refl

b

n

n

, which is not the case.
Thus,

q

m

≠

(

n

↦

refl

b

n

m

)

, and so there is a point of

Ω

m

⁢

(

C

,

c

)

which is unequal to

refl

c

m

.
Hence

C

is not an

m

-type, for any

m

:

N

.

We expect it should also be possible to show that a

universe

𝒰

itself is not an

n

-type for any

n

, using the fact that it contains

higher inductive types

such as

𝕊

n

for all

n

.
However, this has not yet been done.

Title

8.8 Whiteheadâs theorem and Whiteheadâs principle

\metatable

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