Chapter 2 notes

Chapter 2 notes

The definition of identity types, with their

induction

principle, is due to Martin-Löf

[

12

]

.
As mentioned in the notes to

http://

planetmath

.org/node/87533

Chapter 1, our identity types are those that belong to

intensional

type theory

, by contrast with those of

extensional

type theory which have an additional “

reflection

rule” saying that if

p

:

x

=

y

, then in fact

x

≡

y

.
This reflection rule implies that all the higher

groupoid

structure

collapses (see

http://planetmath.org/node/87645

Exercise 2.14), so for nontrivial

homotopy

we must use the intensional version.
One may argue, however, that

homotopy type theory

is in another sense more “extensional” than traditional extensional type theory, because of the function extensionality and univalence axioms.

The proofs of

symmetry

(

inversion

) and

transitivity

(

concatenation

) for equalities are well-known in type theory.
The fact that these make each type into a 1-groupoid (up to homotopy) was exploited in

[

7

]

to give the first “homotopy” style

semantics

for type theory.

The actual homotopical

interpretation

, with identity types as path spaces, and type families as

fibrations

, is due to

[

2

]

, who used the

formalism

of Quillen model categories. An interpretation in (strict)

∞

-groupoids was also given in the thesis

[

13

]

.
For a construction of

all

the higher

operations

and coherences of an

∞

-groupoid in type theory, see

[

14

]

and

[

3

]

.

Operations such as

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

P

⁢

(

p

,

–

)

and

𝖺𝗉

f

, and one good notion of

equivalence

, were first studied extensively in type theory by Voevodsky, using the proof assistant

Coq

.
Subsequently, many other

equivalent

definitions of equivalence have been found, which are compared in

http://planetmath.org/node/87577

Chapter 4.

The “computational” interpretation of identity types, transport, and so on described in §2.5 (

http://planetmath.org/25thehighergroupoidstructureoftypeformers

) has been emphasized by

[

10

]

.
They also described a “1-truncated” type theory (see

http://planetmath.org/node/87580

Chapter 7) in which these rules are judgmental equalities.
The possibility of extending this to the full untruncated theory is a subject of current research.

The naive form of function extensionality which says that “if two

functions

are

pointwise

equal, then they are equal” is a common axiom in type theory, going all the way back to

[

15

]

.
Some stronger forms of function extensionality were considered in

[

5

]

.
The version we have used, which identifies the identity types of function types up to equivalence, was first studied by Voevodsky, who also proved that it is implied by the naive version (and by univalence; see §4.9 (

http://planetmath.org/49univalenceimpliesfunctionextensionality

)).

The univalence axiom is also due to Voevodsky.
It was originally motivated by semantic considerations; see

[

8

]

.

In the type theory we are using in this book, function extensionality and univalence have to be assumed as axioms, i.e.

elements

asserted to belong to some type but not constructed according to the rules for that type.
While serviceable, this has a few drawbacks.
For

instance

, type theory is formally better-behaved if we can base it entirely on rules rather than asserting axioms.
It is also sometimes inconvenient that the

theorems

of §2.6 (

http://planetmath.org/26cartesianproducttypes

) to §2.13 (

http://planetmath.org/213naturalnumbers

) are only propositional equalities (paths) or equivalences, since then we must explicitly mention whenever we pass

back and forth

across them.
One direction of current research in homotopy type theory is to describe a type system in which these rules are

judgmental

equalities, solving both of these problems at once.
So far this has only been done in some simple cases, although preliminary results such as

[

10

]

are promising.
There are also other potential ways to introduce univalence and function extensionality into a type theory, such as having a sufficiently powerful notion of “higher

quotients

” or “higher inductive-recursive types”.

The simple

conclusions

in §2.12 (

http://planetmath.org/212coproducts

) to §2.13 (

http://planetmath.org/213naturalnumbers

) such as “

𝗂𝗇𝗅

and

𝗂𝗇𝗋

are

injective

and

disjoint

” are well-known in type theory, and the construction of the function

𝖾𝗇𝖼𝗈𝖽𝖾

is the usual way to prove them.
The more refined approach we have described, which characterizes the entire identity type of a positive type (up to equivalence), is a more recent development; see e.g.

[

11

]

.

The type-theoretic

axiom of choice

(2.15.6) (

http://planetmath.org/215universalproperties#S0.E4

) was noticed in William Howard’s original paper

[

6

]

on the propositions-as-types correspondence, and was studied further by Martin-Löf with the

introduction

of his dependent type theory. It is mentioned as a “distributivity law” in

Bourbaki

’s

set theory

[

4

]

.

For a more comprehensive (and formalized) discussion of

pullbacks

and more general homotopy limits in homotopy type theory, see

[

1

]

.
Limits of

diagrams

over directed graphs are the easiest general sort of limit to formalize; the problem with diagrams over

categories

(or more generally

(

∞

,

1

)

-categories)
is that in general, infinitely many coherence conditions are

involved in

the notion of (homotopy coherent) diagram.
Resolving this problem is an important open question in homotopy type theory.

References

1

Jeremy Avigad and Krzysztof Kapulkin and Peter LeFanu Lumsdaine. Homotopy limits in Coq, 2013.

\href

http://arxiv.org/abs/1304.0680/arXiv:1304.0680.

2

Steve Awodey and Michael A. Warren. Homotopy theoretic models of identity types.

Mathematical Proceedings of the Cambridge Philosophical Society

, 146:45–55 2009

3

van den Berg,Benno and Garner,Richard. Types are weak

ω

-groupoids.

Proceedings of the

London Mathematical Society

, 102:370–394 2011,Bdsk-Url-1

4

Nicolas Bourbaki,

Theory of Sets

Hermann,Paris,1968

5

Garner,Richard. On the strength of dependent products in the type theory of Martin-Löf.

Annals of Pure and Applied Logic

, 160:1–12 2009,Bdsk-Url-1

6

Howard,William A.. The formulae-as-types notion of construction. In

To H.B. Curry: Essays on

Combinatory Logic

,

Lambda Calculus

and Formalism

,pages 479–490. Academic Press, 1980.

7

Hofmann,Martin and Streicher,Thomas. The groupoid interpretation of type theory. In

Twenty-five years of constructive type theory (Venice,1995)

,volume 36 of Oxford Logic Guides, pages 83–111. Oxford University Press, 1998.

8

Chris Kapulkin and Peter LeFanu Lumsdaine and Vladimir Voevodsky. The Simplicial Model of

Univalent Foundations

, 2012.

\href

http://arxiv.org/abs/1211.2851/arXiv:1211.2851.

9

Licata,Daniel R. and Harper,Robert. Canonicity for 2-dimensional type theory. In emphProceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pages 337–348ACM, .

10

Licata,Daniel R. and Harper,Robert. Canonicity for 2-dimensional type theory. In emphProceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pages 337–348ACM, .

11

Daniel R. Licata and Michael Shulman. Calculating the

Fundamental Group

of the Circle in Homotopy Type Theory. In emphLICS 2013: Proceedings of the Twenty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science, .

12

Martin-Löf,Per. An intuitionistic

theory of types

. In

Twenty-five years of constructive type theory (Venice,1995)

,volume 36 of Oxford Logic Guides, pages 127–172. Oxford University Press, 1998.

13

Michael A. Warren.

Homotopy Theoretic Aspects of Constructive Type Theory

. PhD thesis, Carnegie Mellon University, 2008.

14

Lumsdaine,Peter LeFanu. Weak omega-categories from intensional type theory.

Typed lambda calculi and applications

, 6:1–19 2010,Bdsk-Url-1

15

Whitehead,Alfred North and Russell,Bertrand,

Principia mathematica,3 vol.s

Cambridge University Press,1910–1913; Second edition,1925–1927

Title

Chapter 2 notes

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