Open problems

Open problems

For those interested in contributing to this new branch of mathematics, it may be encouraging to know that there are many interesting open questions.

Perhaps the most pressing of them is the “

constructivity

” of the Univalence Axiom, posed by Voevodsky in

[

1

]

.
The basic system of

type theory

follows the

structure

of Gentzen’s

natural deduction

.

Logical connectives

are defined by their

introduction

rules, and have elimination rules justified by computation rules. Following this pattern, and using Tait’s computability method, originally designed to analyse Gödel’s Dialectica interpretation, one can show the property of

normalization

for type theory. This in turn implies important properties such as decidability of type-checking (a crucial property since type-checking corresponds to proof-checking, and one can argue that we should be able to “recognize a proof when we see one”), and the so-called “canonicity property” that any closed term of the type of

natural numbers

reduces to a numeral. This last property, and the

uniform structure

of introduction/elimination rules, are lost when one extends type theory with an axiom, such as the axiom of function extensionality, or the univalence axiom. Voevodsky has formulated a precise mathematical conjecture

connected

to this question of canonicity for type theory extended with the axiom of Univalence: given a closed term of the type of natural numbers, is it always possible to find a numeral and a proof that this term is equal to this numeral, where this proof of equality may itself use the univalence axiom? More generally, an important issue is whether it is possible to provide a constructive justification of the univalence axiom.
What about if one adds other homotopically motivated constructions, like

higher inductive types

?
These questions remain open at the present time, although methods are currently being developed to try to find answers.

Another basic issue is the difficulty of working with types, such as the natural numbers, that are essentially sets (i.e.,

discrete spaces

),
containing only

trivial

paths.
At present,

homotopy type theory

can really only characterize spaces up to

homotopy equivalence

, which means that these “discrete spaces” may only be

homotopy equivalent

to discrete spaces.
Type-theoretically, this means there are many paths that are equal to

reflexivity

, but not

judgmentally

equal to it (see §1.1 (

http://planetmath.org/11typetheoryversussettheory

) for the meaning of “judgmentally”).
While this homotopy-invariance has advantages, these “meaningless”

identity

terms do introduce needless complications into

arguments

and constructions, so it would be convenient to have a systematic way of eliminating or collapsing them.

A more specialized, but no less important, problem is the

relation

between homotopy type theory and the research on

higher toposes

currently happening at the

intersection

of higher category theory and

homotopy theory

.
There is a growing conviction among those familiar with both subjects that they are intimately connected.
For

instance

, the notion of a

univalent

universe

should coincide with that of an object classifier, while higher inductive types should be an “

elementary

”

reflection

of local presentability.
More generally, homotopy type theory should be the “internal

language

” of

(

∞

,

1

)

-toposes, just as intuitionistic higher-order logic is the internal language of ordinary 1-toposes.
Despite this general consensus, however, details remain to be worked out — in particular, questions of coherence and strictness remain to be addressed — and doing so will undoubtedly lead to further insights into both

concepts

.

But by far the largest field of work to be done is in the ongoing formalization of everyday mathematics in this new system.
Recent successes in formalizing some facts from basic homotopy theory and

category theory

have been encouraging; some of these are described in

http://planetmath.org/node/87582

Chapter 8,

http://planetmath.org/node/87583

Chapter 9.
Obviously, however, much work remains to be done.

The homotopy type theory community maintains a web site and group blog at

\url

http://homotopytypetheory.org, as well as a discussion email list.
Newcomers are always welcome!

References

1

Vladimir Voevodsky. A universe polymorphic type system.

\url

http://uf-ias-2012.wikispaces.com/file/view/Universe+polymorphic+type+sytem.pdf, 2012.

Title

Open problems

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