8. Homotopy theory

8. Homotopy theory

In this chapter, we develop some

homotopy theory

within

type theory

. We
use the

synthetic approach

to homotopy theory introduced in

\cref

cha:basics: Spaces, points, paths, and

homotopies

are basic
notions, which are represented by types and elements of types, particularly
the identity type. The

algebraic structure

of paths and homotopies is
represented by the natural

∞

-

groupoid

structure

on types, which is generated
by the rules for the identity type. Using

higher inductive types

, as
introduced in

\cref

cha:hits, we can describe spaces directly by their

universal properties

.

There are several interesting aspects of this synthetic approach.
First, it combines advantages of concrete models (such as

topological
spaces

or

simplicial sets

)
with advantages of abstract

categorical

frameworks
for homotopy theory (such as Quillen model categories).
On the one hand,
our proofs feel

elementary

, and refer concretely to
points, paths, and homotopies in types. On the other hand, our approach nevertheless abstracts away from
any concrete

presentation

of these objects — for example,

associativity

of path

concatenation

is proved by path

induction

, rather
than by reparametrization of maps

[

0

,

1

]

→

X

or by horn-filling conditions.
Type theory seems to be a very convenient way to study the abstract homotopy theory
of

∞

-groupoids: by using the rules for the identity type, we
can avoid the complicated combinatorics

involved in

many definitions of

∞

-groupoids, and explicate only as much of the
structure as is needed in any particular proof.

The abstract nature of type theory means that our proofs apply automatically in a

variety

of settings.
In particular, as mentioned previously,

homotopy type theory

has one

interpretation

in
Kan simplicial sets,
which is one model for the homotopy theory of

∞

-groupoids. Thus,
our proofs apply to this model, and transferring them along the

geometric
realization

functor

from simplicial sets to topological spaces gives
proofs of corresponding

theorems

in classical homotopy theory.
However, though the details are work in progress, we can also
interpret type theory in a wide variety of other

categories

that look like the category of

∞

-groupoids, such as

(

∞

,

1

)

-toposes.
Thus, proving a result in type theory will show
that it holds in these settings as well.
This sort of extra generality is well-known as a property of ordinary categorical logic:

univalent foundations

extends it to homotopy theory as well.

Second, our synthetic approach has suggested new type-theoretic
methods and proofs. Some of our proofs are fairly
direct transcriptions of classical proofs. Others have a more
type-theoretic feel, and consist mainly of calculations with

∞

-groupoid

operations

, in a style that is very

similar

to how
computer scientists use type theory to reason about computer programs.
One thing that seems to have permitted these new proofs is the fact that type theory
emphasizes different aspects of homotopy theory than other approaches:
while tools like path induction and the universal properties of higher
inductives are available in a setting like Kan simplicial sets, type
theory elevates their importance, because they are the

only

primitive

tools available for working with these types. Focusing on
these tools had led to new descriptions of familiar constructions such
as the

universal cover

of the circle and the

Hopf fibration

, using just the

recursion

principles for higher inductive types. These descriptions are
very direct, and many of the proofs in this chapter involve
computational calculations with such

fibrations

.
Another new aspect of our proofs is that they are constructive (assuming
univalence and higher inductives types are constructive); we describe an
application of this to homotopy groups of spheres in

\cref

sec:moreresults.

Third, our synthetic approach is very amenable to computer-checked
proofs in proof assistants such as

Coq

and

Agda

.
Almost all of the proofs
described in this chapter have been computer-checked, and many of these
proofs were first given in a proof assistant, and then “unformalized”
for this book. The computer-checked proofs are

comparable

in length and
effort to the informal proofs presented here, and in some cases they are
even shorter and easier to do.

Before turning to the presentation of our results, we briefly review some
basic

concepts

and theorems from homotopy theory for the benefit of the reader who is not familiar with them.
We also give an overview of the
results proved in this chapter.

Homotopy theory is a branch of

algebraic topology

, and uses tools from

abstract algebra

,
such as

group theory

, to investigate properties of spaces. One question
homotopy theorists investigate is how to tell whether two spaces are the
same, where “the same” means

homotopy equivalence

(

continuous
maps

back and forth

that compose to the

identity

up to homotopy—this
gives the opportunity to “correct” maps that don’t exactly compose to
the identity). One common way to tell whether two spaces are the same
is to calculate

algebraic invariants

associated with a space,
which include its

homotopy groups

and

homology

and

cohomology groups

.

Equivalent

spaces have

isomorphic

homotopy/(co)

homology groups

, so if two spaces have different groups,
then they are not equivalent. Thus, these algebraic invariants provide
global information about a space, which can be used to tell spaces
apart, and

complements

the local information provided by notions such as
continuity. For example, the torus locally looks like the

2

-sphere, but it
has a global

difference

, because it as a hole in it, and this difference
is visible in the homotopy groups of these two spaces.

The simplest example of a homotopy group is the

fundamental group

of a space, which is written

π

1

(

X

,

x

0

): Given a space

X

and a
point

x

0

in it, one can make a group whose

elements

are loops at

x

0

(

continuous

paths from

x

0

to

x

0

), considered up to homotopy, with the

group operations

given by the identity path (standing still), path
concatenation, and path reversal. For example, the fundamental group of
the

2

-sphere is

trivial

, but the fundamental group of the torus is not,
which shows that the sphere and the torus are not homotopy equivalent.
The intuition is that every loop on the sphere is

homotopic

to the
identity, because its inside can be filled in. In contrast, a loop on
the torus that goes through the donut’s hole is not homotopic to the
identity, so there are non-trivial elements in the fundamental group.

The

higher homotopy groups

provide additional information about a
space. Fix a point

x

0

in

X

, and consider the

constant

path

𝗋𝖾𝖿𝗅

x

0

. Then the

homotopy classes

of homotopies between

𝗋𝖾𝖿𝗅

x

0

and itself form a group

π

2

⁢

(

X

,

x

0

)

, which tells us something about
the two-dimensional structure of the space. Then

π

3

(

X

,

x

0

) is the
group of homotopy classes of homotopies between homotopies, and so on.
One of the basic problems of algebraic topology is

calculating the homotopy groups of a space

X

, which means
giving a group isomorphism between

π

k

⁢

(

X

,

x

0

)

and some more direct
description of a group (e.g., by a multiplication table or
presentation). Somewhat surprisingly, this is a very difficult
question, even for spaces as simple as the spheres. As can be seen from

\cref

tab:homotopy-groups-of-spheres, some patterns emerge in the
higher homotopy groups of spheres, but there is no general

formula

, and
many homotopy groups of spheres are currently still

unknown

.

\definecolor

xA

\OPTcolormodel

\OPTcolxA

\definecolor

xB

\OPTcolormodel

\OPTcolxB

\definecolor

xC

\OPTcolormodel

\OPTcolxC

\definecolor

xD

\OPTcolormodel

\OPTcolxD

\definecolor

xE

\OPTcolormodel

\OPTcolxE

\definecolor

xF

\OPTcolormodel

\OPTcolxF

\definecolor

xG

\OPTcolormodel

\OPTcolxG

\definecolor

xH

\OPTcolormodel

\OPTcolxH

\definecolor

xI

\OPTcolormodel

\OPTcolxI

\definecolor

xJ

\OPTcolormodel

\OPTcolxJ

\definecolor

xK

\OPTcolormodel

\OPTcolxK

\definecolor

xL

\OPTcolormodel

\OPTcolxL

\definecolor

xM

\OPTcolormodel

\OPTcolxM

\toprule

𝕊

0

𝕊

1

𝕊

2

𝕊

3

𝕊

4

𝕊

5

𝕊

6

𝕊

7

𝕊

8

\addlinespace

[3pt]

\midrule

π

1

0

ℤ

\colorbox

xG

0

\colorbox

xF

0

\colorbox

xE

0

\colorbox

xD

0

\colorbox

xC

0

\colorbox

xB

0

\colorbox

xA

0

\addlinespace

[3pt]

π

2

0

0

\colorbox

xH

ℤ

\colorbox

xG

0

\colorbox

xF

0

\colorbox

xE

0

\colorbox

xD

0

\colorbox

xC

0

\colorbox

xB

0

\addlinespace

[3pt]

π

3

0

0

ℤ

\colorbox

xH

ℤ

\colorbox

xG

0

\colorbox

xF

0

\colorbox

xE

0

\colorbox

xD

0

\colorbox

xC

0

\addlinespace

[3pt]

π

4

0

0

ℤ

2

\colorbox

xI

ℤ

2

\colorbox

xH

ℤ

\colorbox

xG

0

\colorbox

xF

0

\colorbox

xE

0

\colorbox

xD

0

\addlinespace

[3pt]

π

5

0

0

ℤ

2

ℤ

2

\colorbox

xI

ℤ

2

\colorbox

xH

ℤ

\colorbox

xG

0

\colorbox

xF

0

\colorbox

xE

0

\addlinespace

[3pt]

π

6

0

0

ℤ

12

ℤ

12

\colorbox

xJ

ℤ

2

\colorbox

xI

ℤ

2

\colorbox

xH

ℤ

\colorbox

xG

0

\colorbox

xF

0

\addlinespace

[3pt]

π

7

0

0

ℤ

2

ℤ

2

ℤ

×

ℤ

12

\colorbox

xJ

ℤ

2

\colorbox

xI

ℤ

2

\colorbox

xH

ℤ

\colorbox

xG

0

\addlinespace

[3pt]

π

8

0

0

ℤ

2

ℤ

2

ℤ

2

2

\colorbox

xK

ℤ

24

\colorbox

xJ

ℤ

2

\colorbox

xI

ℤ

2

\colorbox

xH

ℤ

\addlinespace

[3pt]

π

9

0

0

ℤ

3

ℤ

3

ℤ

2

2

ℤ

2

\colorbox

xK

ℤ

24

\colorbox

xJ

ℤ

2

\colorbox

xI

ℤ

2

\addlinespace

[3pt]

π

10

0

0

ℤ

15

ℤ

15

ℤ

24

×

ℤ

3

ℤ

2

\colorbox

xL

0

\colorbox

xK

ℤ

24

\colorbox

xJ

ℤ

2

\addlinespace

[3pt]

π

11

0

0

ℤ

2

ℤ

2

ℤ

15

ℤ

2

ℤ

\colorbox

xL

0

\colorbox

xK

ℤ

24

\addlinespace

[3pt]

π

12

0

0

ℤ

2

2

ℤ

2

2

ℤ

2

ℤ

30

ℤ

2

\colorbox

xM

0

\colorbox

xL

0

\addlinespace

[3pt]

π

13

0

0

ℤ

12

×

ℤ

2

ℤ

12

×

ℤ

2

ℤ

2

3

ℤ

2

ℤ

60

ℤ

2

\colorbox

xM

0

\addlinespace

[3pt]

\bottomrule

Table 1:

Homotopy groups of spheres

[

wikipedia-groups

]

.
The

k

th

homotopy group

π

k

of the

n

-dimensional sphere

𝕊

n

is isomorphic to the group listed in each entry, where

ℤ

is
the

additive group

of integers, and

ℤ

m

is the

cyclic group

of order

m

.

One way of understanding this complexity is through the correspondence
between spaces and

∞

-groupoids
introduced in

\cref

cha:basics.
As discussed in

\cref

sec:circle, the 2-sphere is presented by a higher
inductive type with one point and one 2-dimensional loop. Thus, one
might wonder why

π

3

⁢

(

𝕊

2

)

is

ℤ

, when the type

𝕊

2

has no

generators

creating 3-dimensional cells. It turns out that the
generating element of

π

3

⁢

(

𝕊

2

)

is constructed using the

interchange law

described in the proof of

\cref

thm:EckmannHilton: the
algebraic structure of an

∞

-groupoid includes non-trivial
interactions between levels, and these interactions create elements of
higher homotopy groups.

Type theory provides a natural setting for investigating this structure, as
we can easily define the higher homotopy groups. Recall from

\cref

def:loopspace that for

n

:

ℕ

, the

n

-fold iterated loop space of a pointed type

(

A

,

a

)

is defined
recursively by:

Ω

0

⁢

(

A

,

a

)

=

(

A

,

a

)

Ω

n

+

1

⁢

(

A

,

a

)

=

Ω

n

⁢

(

Ω

⁢

(

A

,

a

)

)

.

This gives a

space

(i.e. a type) of

n

-dimensional loops, which itself has
higher homotopies. We obtain the set of

n

-dimensional loops by
truncation (this was also defined as an example in

\autoref

sec:free-algebras):

Definition 8.0.1

(Homotopy Groups).

Given

n

≥

1

and

(

A

,

a

)

a pointed type, we define the

homotopy groups

of

A

at

a

by

π

n

(

A

,

a

)

:

≡

∥

Ω

n

(

A

,

a

)

∥

0

Since

n

≥

1

, the path concatenation and

inversion operations

on

Ω

n

⁢

(

A

)

induce operations on

π

n

⁢

(

A

)

making it into a group in
a straightforward way. If

n

≥

2

, then the group

π

n

⁢

(

A

)

is

abelian

, by the Eckmann–Hilton

argument

(

\autoref

thm:EckmannHilton).
It is convenient to also write

π

0

(

A

)

:

≡

∥

A

∥

0

,
but this case behaves somewhat differently: not only is it not a group,
it is defined without reference to any

basepoint

in

A

.

This definition is a suitable one for investigating homotopy groups
because the (higher) inductive definition of a type

X

presents

X

as
a free type, analogous to a free

∞

-groupoid,
and this
presentation

determines

but does not

explicitly describe

the higher identity types of

X

. The identity types are populated by
both the generators (

𝗅𝗈𝗈𝗉

, for the circle) and the results of applying to them all of the groupoid
operations (identity,

composition

,

inverses

, associativity, interchange,
…). Thus, the higher-inductive presentation of a space allows us
to pose the question “what does the identity type of

X

really turn out
to be?” though it can take some significant mathematics to answer it.
This is a higher-dimensional

generalization

of a familiar fact in type
theory: characterizing the identity type of

X

can take some work,
even if

X

is an ordinary inductive type, such as the

natural numbers

or

booleans

. For example, the theorem that

0

𝟐

is different
from

1

𝟐

does not follow immediately from the definition;
see

\autoref

sec:compute-coprod.

The univalence axiom plays an essential role in calculating homotopy
groups (without univalence, type theory is

compatible

with an
interpretation where all paths, including, for example, the loop on the
circle, are

reflexivity

). We will see this in the calculation of the
fundamental group of the circle below: the map from

Ω

⁢

(

𝕊

1

)

to

ℤ

is defined by

mapping

a loop on the circle to an

automorphism

of the set

ℤ

, so that, for example,

𝗅𝗈𝗈𝗉

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

-

1

is sent to

𝗌𝗎𝖼𝖼𝖾𝗌𝗌𝗈𝗋

⁢

\centerdot

⁢

𝗉𝗋𝖾𝖽𝖾𝖼𝖾𝗌𝗌𝗈𝗋

(where

𝗌𝗎𝖼𝖼𝖾𝗌𝗌𝗈𝗋

and

𝗉𝗋𝖾𝖽𝖾𝖼𝖾𝗌𝗌𝗈𝗋

are automorphisms of

ℤ

viewed, by univalence, as paths in the

universe

), and then applying
the automorphism to 0. Univalence produces non-trivial paths in the
universe, and this is used to extract information from paths in higher
inductive types.

In this chapter, we first calculate some homotopy groups of spheres,
including

π

k

⁢

(

𝕊

1

)

(

\autoref

sec:pi1-s1-intro),

π

k

⁢

(

𝕊

n

)

for

k

<

n

(

\autoref

sec:conn-susp,

\autoref

sec:pik-le-n),

π

2

⁢

(

𝕊

2

)

and

π

3

⁢

(

𝕊

2

)

by
way of the Hopf fibration (

\autoref

sec:hopf) and a long-exact-sequence
argument (

\autoref

sec:long-exact-sequence-homotopy-groups), and

π

n

⁢

(

𝕊

n

)

by way of the

Freudenthal suspension theorem

(

\autoref

sec:freudenthal). Next, we discuss the

van Kampen theorem

(

\autoref

sec:van-kampen), which characterizes the fundamental group of
a

pushout

, and the status of Whitehead’s principle (when is a map that
induces an

equivalence

on all homotopy groups an equivalence?)
(

\autoref

sec:whitehead). Finally, we include brief summaries of
additional results that are not included in the book, such as

π

n

+

1

⁢

(

𝕊

n

)

for

n

≥

3

, the Blakers–Massey theorem, and a
construction of Eilenberg–Mac Lane spaces (

\autoref

sec:moreresults).
Prerequisites for this chapter include

\autoref

cha:typetheory,

\autoref

cha:basics,

\autoref

cha:hits,

\autoref

cha:hlevels as well as parts of

\autoref

cha:logic.

Title

8. Homotopy theory

\metatable

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