2.5 The higher groupoid structure of type formers

2.5 The higher groupoid structure of type formers

In

http://planetmath.org/node/87533

Chapter 1, we introduced many ways to form new types:

cartesian products

,

disjoint unions

, dependent products, dependent sums, etc.
In §2.1 (

http://planetmath.org/21typesarehighergroupoids

),§2.2 (

http://planetmath.org/22functionsarefunctors

),§2.3 (

http://planetmath.org/23typefamiliesarefibrations

), we saw that

all

types in

homotopy type theory

behave like spaces or higher

groupoids

.
Our goal in the rest of the chapter is to make explicit how this higher

structure

behaves in the case of the particular types defined in

http://planetmath.org/node/87533

Chapter 1.

It turns out that for many types

A

, the

equality

types

x

=

A

y

can be characterized, up to

equivalence

, in terms of whatever data was used to construct

A

.
For example, if

A

is a cartesian product

B

×

C

, and

x

≡

(

b

,

c

)

and

y

≡

(

b

′

,

c

′

)

, then we have an equivalence

(

(

b

,

c

)

=

(

b

′

,

c

′

)

)

≃

(

(

b

=

b

′

)

×

(

c

=

c

′

)

)

.

(2.5.1)

In more traditional

language

, two

ordered pairs

are equal just when their components are equal (but the equivalence (

2.5.1

) says rather more than this).
The higher structure of the identity types can also be expressed in terms of these equivalences; for

instance

, concatenating two equalities between pairs corresponds to pairwise

concatenation

.

Similarly, when a type family

P

:

A

→

𝒰

is built up fiberwise using the type forming rules from

http://planetmath.org/node/87533

Chapter 1, the

operation

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

P

⁢

(

p

,

–

)

can be characterized, up to

homotopy

, in terms of the corresponding operations on the data that went into

P

.
For instance, if

P

⁢

(

x

)

≡

B

⁢

(

x

)

×

C

⁢

(

x

)

, then we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

P

⁢

(

p

,

(

b

,

c

)

)

=

(

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

p

,

b

)

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

C

⁢

(

p

,

c

)

)

.

Finally, the type forming rules are also functorial, and if a

function

f

is built from this functoriality, then the operations

𝖺𝗉

f

and

𝖺𝗉𝖽

f

can be computed based on the corresponding ones on the data going into

f

.
For instance, if

g

:

B

→

B

′

and

h

:

C

→

C

′

and we define

f

:

B

×

C

→

B

′

×

C

′

by

f

(

b

,

c

)

:

≡

(

g

(

b

)

,

h

(

c

)

)

, then modulo the equivalence (

2.5.1

), we can identify

𝖺𝗉

f

with “

(

𝖺𝗉

g

,

𝖺𝗉

h

)

”.

The next few

sections

(§2.6 (

http://planetmath.org/26cartesianproducttypes

) to §2.13 (

http://planetmath.org/213naturalnumbers

)) will be devoted to stating and proving

theorems

of this sort for all the basic type forming rules, with one section for each basic type former.
Here we encounter a certain apparent deficiency in currently available

type theories

;
as will become clear in later chapters, it would seem to be more convenient and intuitive if these

characterizations

of identity types, transport, and so on were

judgmental

equalities.
However, in the theory presented in

http://planetmath.org/node/87533

Chapter 1, the identity types are defined uniformly for all types by their

induction

principle, so we cannot “redefine” them to be different things at different types.
Thus, the characterizations for particular types to be discussed in this chapter are, for the most part,

theorems

which we have to discover and prove, if possible.

Actually, the type theory of

http://planetmath.org/node/87533

Chapter 1 is insufficient to prove the desired theorems for two of the type formers:

Π

-types and

universes

.
For this reason, we are forced to introduce axioms into our type theory, in order to make those “theorems” true.
Type-theoretically, an

axiom

(c.f. §1.1 (

http://planetmath.org/11typetheoryversussettheory

)) is an “atomic”

element

that is declared to inhabit some specified type, without there being any rules governing its behavior other than those pertaining to the type it inhabits.

The axiom for

Π

-types (§2.9 (

http://planetmath.org/29pitypesandthefunctionextensionalityaxiom

)) is familiar to type theorists: it is called

function extensionality

, and states (roughly) that if two functions are

homotopic

in the sense of §2.4 (

http://planetmath.org/24homotopiesandequivalences

), then they are equal.
The axiom for universes (§2.10 (

http://planetmath.org/210universesandtheunivalenceaxiom

)), however, is a new contribution of homotopy type theory due to Voevodsky: it is called the

univalence axiom

, and states (roughly) that if two types are

equivalent

in the sense of §2.4 (

http://planetmath.org/24homotopiesandequivalences

), then they are equal.
We have already remarked on this axiom in the

introduction

; it will play a very important role in this book.

1

1

We have chosen to introduce these principles as axioms, but there are potentially other ways to formulate a type theory in which they hold.
See the Notes to this chapter.

It is important to note that not

all

identity types can be “determined” by induction over the construction of types.
Counterexamples include most nontrivial

higher inductive types

(see

http://planetmath.org/node/87579

Chapter 6,

http://planetmath.org/node/87582

Chapter 8).
For instance, calculating the identity types of the types

𝕊

n

(see §6.4 (

http://planetmath.org/64circlesandspheres

)) is equivalent to calculating the

higher homotopy groups

of spheres, a deep and important field of research in

algebraic topology

.

Title

2.5 The higher groupoid structure of type formers

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