2. Homotopy Type Theory

2. Homotopy Type Theory

The

central

new idea in

homotopy type theory

is that types can be regarded as
spaces in

homotopy theory

, or higher-dimensional

groupoids

in

category
theory

.

We begin with a brief summary of the

connection

between homotopy theory
and higher-dimensional category theory.
In classical homotopy theory, a space

X

is a set of points equipped
with a

topology

,

and a path between points

x

and

y

is represented by
a

continuous map

p

:

[

0

,

1

]

→

X

, where

p

⁢

(

0

)

=

x

and

p

⁢

(

1

)

=

y

.

This

function

can be thought of as giving a point in

X

at each
“moment in time”. For many purposes, strict equality of paths
(meaning,

pointwise

equal functions) is too fine a notion. For example,
one can define

operations

of path

concatenation

(if

p

is a path from

x

to

y

and

q

is a path from

y

to

z

, then the concatenation

p

⁢

\centerdot

⁢

q

is a path from

x

to

z

) and

inverses

(

p

-

1

is a path
from

y

to

x

). However, there are natural equations between these
operations that do not hold for strict equality: for example, the path

p

⁢

\centerdot

⁢

p

-

1

(which walks from

x

to

y

, and then back along the
same route, as time goes from

0

to

1

) is not strictly equal to the

identity

path (which stays still at

x

at all times).

The remedy is to consider a

coarser

notion of equality of paths called

homotopy

.

A homotopy between a pair of continuous maps

f

:

X

1

→

X

2

and

g

:

X

1

→

X

2

is a continuous map

H

:

X

1

×

[

0

,

1

]

→

X

2

satisfying

H

⁢

(

x

,

0

)

=

f

⁢

(

x

)

and

H

⁢

(

x

,

1

)

=

g

⁢

(

x

)

. In the specific case of paths

p

and

q

from

x

to

y

, a homotopy is a
continuous map

H

:

[

0

,

1

]

×

[

0

,

1

]

→

X

such that

H

⁢

(

s

,

0

)

=

p

⁢

(

s

)

and

H

⁢

(

s

,

1

)

=

q

⁢

(

s

)

for all

s

∈

[

0

,

1

]

.
In this case we require also that

H

⁢

(

0

,

t

)

=

x

and

H

⁢

(

1

,

t

)

=

y

for all

t

∈

[

0

,

1

]

,
so that for each

t

the function

H

⁢

(

–

,

t

)

is again a path from

x

to

y

;
a homotopy of this sort is said to be

endpoint-preserving

or

rel

endpoints

.
Such a homotopy is the
image in

X

of a square that fills in the space between

p

and

q

,
which can be thought of as a “continuous deformation” between

p

and

q

, or a 2-dimensional

path between paths

.

For example, because

p

⁢

\centerdot

⁢

p

-

1

walks out and back along the same route, you know that
you can continuously shrink

p

⁢

\centerdot

⁢

p

-

1

down to the identity
path—it won’t, for example, get snagged around a hole in the space.
Homotopy is an

equivalence relation

, and operations such as
concatenation, inverses, etc., respect it. Moreover, the

homotopy
equivalence

classes of loops at some point

x

0

(where two loops

p

and

q

are equated when there is a

based

homotopy between them,
which is a homotopy

H

as above that additionally

satisfies

H

⁢

(

0

,

t

)

=

H

⁢

(

1

,

t

)

=

x

0

for all

t

) form a group called the

fundamental
group

. This group is an

algebraic invariant

of a space, which
can be used to investigate whether two spaces are

homotopy
equivalent

(there are continuous maps

back and forth

whose

composites

are

homotopic

to the identity), because

equivalent

spaces have

isomorphic

fundamental groups.

Because homotopies are themselves a kind of 2-dimensional path, there is
a natural notion of 3-dimensional

homotopy between homotopies

,
and then

homotopy between homotopies between homotopies

, and so
on. This

infinite

tower of points, path, homotopies, homotopies between
homotopies, …, equipped with algebraic operations such as the
fundamental group, is an

instance

of an

algebraic structure

called a
(weak)

∞

-groupoid

. An

∞

-groupoid consists of a

collection

of objects, and then a collection of

morphisms

between
objects, and then

morphisms between morphisms

, and so on,
equipped with some

complex

algebraic structure; a morphism at level

k

is called a

k

-morphism

. Morphisms at each level
have identity, composition, and inverse operations, which are weak in
the sense that they satisfy the groupoid laws (

associativity

of
composition, identity is a unit for composition, inverses cancel) only
up to morphisms at the next level, and this weakness gives rise to
further

structure

. For example, because associativity of composition of
morphisms

p

⁢

\centerdot

⁢

(

q

⁢

\centerdot

⁢

r

)

=

(

p

⁢

\centerdot

⁢

q

)

⁢

\centerdot

⁢

r

is itself a
higher-dimensional morphism, one needs an additional operation relating
various proofs of associativity: the various ways to reassociate

p

⁢

\centerdot

⁢

(

q

⁢

\centerdot

⁢

(

r

⁢

\centerdot

⁢

s

)

)

into

(

(

p

⁢

\centerdot

⁢

q

)

⁢

\centerdot

⁢

r

)

⁢

\centerdot

⁢

s

give rise to Mac
Lane’s

pentagon

. Weakness also creates non-trivial interactions between
levels.

Every

topological space

X

has a

fundamental

∞

-groupoid

whose

k

-morphisms are the

k

-dimensional paths in

X

. The weakness of the

∞

-groupoid corresponds directly to the fact that paths form a
group only up to homotopy, with the

(

k

+

1

)

-paths serving as the
homotopies between the

k

-paths. Moreover, the view of a space as an

∞

-groupoid preserves enough aspects of the space to do homotopy theory:
the fundamental

∞

-groupoid construction is

adjoint

to the

geometric realization

of an

∞

-groupoid as a space, and this

adjunction

preserves homotopy theory (this is called the

homotopy
hypothesis/

theorem

,

because whether it is a

hypothesis

or theorem
depends on how you define

∞

-groupoid). For example, you can
easily define the fundamental group of an

∞

-groupoid, and if you
calculate the fundamental group of the fundamental

∞

-groupoid of
a space, it will agree with the classical definition of fundamental
group of that space. Because of this correspondence, homotopy theory
and higher-dimensional category theory are intimately related.

Now, in homotopy type theory each type can be seen to have the structure
of an

∞

-groupoid. Recall that for any type

A

, and any

x

,

y

:

A

,
we have a identity type

x

=

A

y

, also written

𝖨𝖽

A

⁢

(

x

,

y

)

or just

x

=

y

. Logically, we may think of elements of

x

=

y

as evidence
that

x

and

y

are equal, or as identifications of

x

with

y

. Furthermore,

type theory

(unlike, say,

first-order logic

) allows us
to consider such elements of

x

=

A

y

also as

individuals

which
may be the subjects of further

propositions

. Therefore, we can

iterate

the identity type: we can form the type

p

=

(

x

=

A

y

)

q

of identifications between
identifications

p

,

q

, and the type

r

=

(

p

=

(

x

=

A

y

)

q

)

s

, and so on. The structure
of this tower of identity types corresponds precisely to that of the

continuous

paths and (higher) homotopies between them in a space, or an

∞

-groupoid.

Thus, we will frequently refer to an

element

p

:

x

=

A

y

as
a

path

from

x

to

y

; we call

x

its

start point

and

y

its

end point

.

Two paths

p

,

q

:

x

=

A

y

with the same start and end point are said to be

parallel

,

in which case an element

r

:

p

=

(

x

=

A

y

)

q

can
be thought of as a homotopy, or a morphism between morphisms;
we will often refer to it as a

2-path

or a

2-dimensional path

Similarly,

r

=

(

p

=

(

x

=

A

y

)

q

)

s

is the type of

3-dimensional paths

between two parallel 2-dimensional paths, and so on. If the
type

A

is “set-like”, such as

ℕ

, these iterated identity types
will be uninteresting (see §3.1 (

http://planetmath.org/31setsandntypes

)), but in the
general case they can model non-trivial

homotopy types

.

An important

difference

between homotopy type theory and classical homotopy theory is that homotopy type theory provides a

synthetic

description of spaces, in the following sense. Synthetic geometry is

geometry

in the style of Euclid

[

1

]

: one starts from some basic notions (points and lines), constructions (a line connecting any two points), and axioms
(all

right angles

are equal), and deduces

consequences

logically. This is in contrast with

analytic
geometry

, where notions such as points and lines are represented concretely using

cartesian coordinates

in

ℝ

n

—lines are sets of points—and the basic constructions and axioms are derived from this

representation

. While classical homotopy theory is

analytic

(spaces and paths are made of points), homotopy type theory is synthetic: points, paths, and paths between paths are basic, indivisible, primitive notions.

Moreover, one of the amazing things about homotopy type theory is that all of the basic constructions and axioms—all of the
higher groupoid structure—-arises automatically from the

induction

principle for identity types.
Recall from §1.12 (

http://planetmath.org/112identitytypes

) that this says that if

•

for every

x

,

y

:

A

and every

p

:

x

=

A

y

we have a type

D

⁢

(

x

,

y

,

p

)

, and

•

for every

a

:

A

we have an element

d

⁢

(

a

)

:

D

⁢

(

a

,

a

,

𝗋𝖾𝖿𝗅

a

)

,

then

•

there exists an element

𝗂𝗇𝖽

=

A

⁢

(

D

,

d

,

x

,

y

,

p

)

:

D

⁢

(

x

,

y

,

p

)

for

every

two elements

x

,

y

:

A

and

p

:

x

=

A

y

, such that

𝗂𝗇𝖽

=

A

⁢

(

D

,

d

,

a

,

a

,

𝗋𝖾𝖿𝗅

a

)

≡

d

⁢

(

a

)

.

In other words, given dependent functions

D

:

∏

(

x

,

y

:

A

)

∏

(

p

:

x

=

y

)

𝒰

d

:

∏

a

:

A

D

⁢

(

a

,

a

,

𝗋𝖾𝖿𝗅

a

)

there is a dependent function

𝗂𝗇𝖽

=

A

⁢

(

D

,

d

)

:

∏

(

x

,

y

:

A

)

∏

(

p

:

x

=

y

)

D

⁢

(

x

,

y

,

p

)

such that

𝗂𝗇𝖽

=

A

⁢

(

D

,

d

,

a

,

a

,

𝗋𝖾𝖿𝗅

a

)

≡

d

⁢

(

a

)

(2.0.1)

for every

a

:

A

.
Usually, every time we apply this induction rule we will either not care about the specific function being defined, or we will immediately give it a different name.

Informally, the induction principle for identity types says that if we want to

construct

an object (or prove a statement) which depends on an inhabitant

p

:

x

=

A

y

of an identity type, then it suffices to perform the construction (or the proof) in the special case when

x

and

y

are the same (judgmentally) and

p

is the

reflexivity

element

𝗋𝖾𝖿𝗅

x

:

x

=

x

(judgmentally).
When writing informally, we may express this with a phrase such as “by induction, it suffices to assume…”.
This

reduction

to the “reflexivity case” is analogous to the reduction to the “

base case

” and “inductive step” in an ordinary proof by induction on the

natural numbers

, and also to the “left case” and “right case” in a proof by case

analysis

on a

disjoint union

or

disjunction

.

The “conversion rule” (

2.0.1

) is less familiar in the

context

of proof by induction on natural numbers, but there is an analogous notion in the related concept of definition by

recursion

.
If a

sequence

(

a

n

)

n

∈

ℕ

is defined by giving

a

0

and specifying

a

n

+

1

in terms of

a

n

, then in fact the

0

th

term of the resulting sequence

is

the given one, and the given

recurrence relation

relating

a

n

+

1

to

a

n

holds for the resulting sequence.
(This may seem so obvious as to not be worth saying, but if we view a definition by recursion as an algorithm for calculating values of a sequence, then it is precisely the process of executing that algorithm.)
The rule (

2.0.1

) is analogous: it says that if we define an object

f

⁢

(

p

)

for all

p

:

x

=

y

by specifying what the value should be when

p

is

𝗋𝖾𝖿𝗅

x

:

x

=

x

, then the value we specified is in fact the value of

f

⁢

(

𝗋𝖾𝖿𝗅

x

)

.

This induction principle endows each type with the structure of an

∞

-groupoid, and each function between two types the structure of an

∞

-

functor

between two such groupoids. This is interesting from a mathematical point view, because it gives a new way to work with

∞

-groupoids. It is interesting from a type-theoretic point view, because it reveals new operations that are associated with each type and function. In the

remainder

of this chapter, we begin to explore this structure.

References

1

Euclid,

Elements,Vols. 1–13

Elsevier,300 BC

Title

2. Homotopy Type Theory

\metatable

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