nLab higher inductive type

Idea

Higher inductive types (HITs) are a generalization of inductive types which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated identity types or bridge types. See the Narya documentation for the use of bridge constructors in HITs.

HITs can use identity types $\mathrm{Id}_{I, A}(\overline{x})$ or bridge types $\mathrm{Br}_{I, A}(\overline{x})$ of any arity $I$. The arity most commonly used in HITs is binary, where identity types and bridge types are denoted respectively as

$$\mathrm{Id}_A(x, y) \coloneqq \mathrm{Id}_{\mathbb{2}, A}(\mathrm{rec}_{\mathbb{2}, A}(x, y)) \qquad \mathrm{Br}_A(x, y) \coloneqq \mathrm{Br}_{\mathbb{2}, A}(\mathrm{rec}_{\mathbb{2}, A}(x, y))$$

and where $\mathbb{2}$ is the boolean type and $\mathrm{rec}_{\mathbb{2}, A}:A \times A \to (\mathbb{2} \to A)$ is the function defined in the recursion principle of the boolean type. Two examples of a HIT which uses identity types with an arity $I$ that is not necessarily binary are the walking identification and the walking bridge of arity $I$.

Either way, the iterated type families used (identity types or bridge types) need to have some notion of function application $\mathrm{ap}$ to identifications or bridges for the recursion principle of the HIT and dependent function application $\mathrm{apd}$ to identifications or bridges for the induction principle of the HIT.

While HITs are already useful in set-level type theory, they are most useful and powerful in homotopy type theory, where they allow the construction of cell complexes, homotopy colimits, truncations, localizations, and many other objects from classical homotopy theory.

Defining what a HIT is “in general” is an open research problem. One mostly precise proposal may be found in [Shulman & Lumsdaine (2016)]. A more syntactic description of a class of HITs may be found in [Brunerie (2016), Vezzosi, Mörtberg & Abel (2019)]. A solution to this problem should determine how to define the concept of an elementary (∞,1)-topos.

Examples

All higher inductive types described below are given together with some pseudo-Coq code, which would implement that HIT if Coq supported HITs natively.

The circle

The circle type is:

Inductive circle : Type :=
| base : circle
| loop : base == base.

Using the univalence axiom, one can prove that the loop space base == base of the circle type is equivalent to the integers.

(Licata & Shulman 13, Bezem, Buchholtz, Grayson & Shulman 19).

The interval

The homotopy type of the interval can be encoded as

Inductive interval : Type :=
| zero : interval
| one : interval
| segment : zero == one.

See interval type. The interval can be proven to be contractible. On the other hand, if the constructors zero and one satisfy their elimination rules definitionally, then the existence of an interval type implies function extensionality; see this blog post. The interval can be defined as the -1-truncation of the booleans; see here:

Inductive interval : Type :=
| inj : boolean -> interval
| contr0 : forall (p q : interval) p == q

Or more directly:

Inductive interval : Type :=
| zero : interval
| one : interval
| contr0 : forall (p q : interval) p == q

The 2-sphere

Similarly the homotopy type of the 2-dimensional sphere

Inductive sphere2 : Type :=
| base2 : sphere2
| surf2 : idpath base2 == idpath base2.

Suspension

Inductive susp (X : Type) : Type :=
| north : susp X
| south : susp X
| merid : X -> north == south.

This is the unpointed suspension. It is also possible to define the pointed suspension. Using either one, we can define the $n$-sphere by induction on $n$, since $S^{n+1}$ is the suspension of $S^n$.

Mapping cylinders

The construction of mapping cylinders is given by

Inductive cyl {X Y : Type} (f : X -> Y) : Y -> Type :=
| cyl_base : forall y:Y, cyl f y
| cyl_top : forall x:X, cyl f (f x)
| cyl_seg : forall x:X, cyl_top x == cyl_base (f x).

Using this construction, one can define a (cofibration, trivial fibration) weak factorization system for types.

Truncation

Inductive is_inhab (A : Type) : Type :=
| inhab : A -> is_inhab A
| inhab_path : forall (x y: is_inhab A), x == y.

This is the (-1)-truncation into h-propositions. One can prove that is_inhab A is always a proposition (i.e. $(-1)$-truncated) and that it is the reflection of $A$ into propositions. More generally, one can construct the (effective epi, mono) factorization system by applying is_inhab fiberwise to a fibration.

Similarly, we have the 0-truncation into h-sets:

Inductive pi0 (X:Type) : Type :=
| cpnt : X -> pi0 X
| pi0_axiomK : forall (l : Circle -> pi0 X), refl (l base) == map l loop.

We can similarly define $n$-truncation for any $n$, and we should be able to define it inductively for all $n$ at once as well.

See at n-truncation modality.

Quotients of types

The quotient of a pure or Type-valued equivalence relation:

Inductive quotient (A : Type) (R : A -> A -> Type) : Type :=
| proj : A -> quotient A R
| relate : forall (x y : A), R x y -> proj x == proj y

This definition is translated into Coq from the Cubical Agda library.

Disjunctions

The disjunction of two types $A$ and $B$, yielding an hProp:

Inductive disjunction (A B:Type) : Type :=
| inl : A -> disjunction A B
| inr : B -> disjunction A B.
| contr0 : forall (p q : disjunction A B) p == q

Existential quantifiers

The existential quantifier of a type $A$ and a type family $B:A \to Type$, yielding an hProp:

Inductive existquant (A:Type) (B:A->Type) : Type :=
| exist : forall (x:A), B x -> existquant A B
| contr0 : forall (p q : existquant A B) p == q

Quotients of sets

The quotient of an hProp-value equivalence relation, yielding an hSet (a 0-truncated type):

Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
| proj : A -> quotient A R
| relate : forall (x y : A), R x y -> proj x == proj y
| contr1 : forall (x y : quotient A R) (p q : x == y), p == q.

This is already interesting in extensional type theory, where quotient types are not always included. For more general homotopical quotients of “internal groupoids” as in the (∞,1)-Giraud theorem, we first need a good definition of what such an internal groupoid is.

Quotient inductive types

A quotient inductive type is a higher inductive type that includes a “0-truncation” constructor such as contr1 for a set-quotient. Many of these are useful in set-based mathematics; in addition to colimits in Set, they can be used to construct free algebras and colimits of algebras of various sorts. Quotient inductive-inductive types are used to construct sets with propositional relations and various countable completions of structures. Examples can be found at quotient inductive type, including:

• multiple definitions of the integers
• polynomial rings
• Sierpinski space
• the cumulative hierarchy
• the Cauchy real numbers
• partial map classifiers
• internal type theory

FinSet

Since FinSet is the initial 2-rig, one should be able to construct it as a higher inductive type with a 1-truncation constructor.

Integers

A definition of the set of integers as a higher inductive type.

Inductive int : Type :=
| zero : int
| succ : int -> int
| pred1 : int -> int
| pred2 : int -> int
| sec : forall (x : int) pred1 succ x == x
| ret : forall (x : int) succ pred2 x == x

Other

• circle type

• sphere type

• cone type

• square type

• pushout type

• suspension type

• join type

• wedge sum type

• smash product type

• localization of a type at a family of functions

• Eilenberg-MacLane space type

• spectrification of a sequential spectrum type

• James construction type

• Rezk completion

• dependent pushout type

Semantics

See (Lumsdaine-Shulman17).

Related concepts

• inductive type, initial algebra of an endofunctor

• higher inductive type, initial algebra of a presentable ∞-monad

• quotient inductive type

• higher coinductive type

References

Textbook account:

• Univalent Foundations Project, §6 of: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

Expositions:

• Mike Shulman, Homotopy type theory IV (blog entry)

• Peter LeFanu Lumsdaine, Higher inductive types, a tour of the menagerie (2011) [blog post]

• Peter LeFanu Lumsdaine, Higher Inductive Types: The circle and friends, axiomatically (2011) [pdf, pdf]

• Kajetan Söhnen, Higher Inductive Types in Homotopy Type Theory, Munich (2018) [pdf, pdf]

Details on the categorical semantics of HITs:

• Peter LeFanu Lumsdaine, Mike Shulman, Semantics of higher inductive types, Math. Proc. Camb. Phil. Soc. 169 (2020) 159-208 [arXiv:1705.07088, talk slides pdf, doi:10.1017/S030500411900015X]

with precursors in

• Mike Shulman, Peter LeFanu Lumsdaine, Semantics of higher inductive types (2012) [pdf]

• Mike Shulman, Peter LeFanu Lumsdaine, Semantics and syntax of higher inductive types (2016) [slides]

Discussion of HITs which arise as homotopy-initial algebras of an endofunctor:

• Kristina Sojakova, Higher Inductive Types as Homotopy-Initial Algebras, ACM SIGPLAN Notices 50 1 (2015) 31–42 [arXiv:1402.0761, doi:10.1145/2775051.2676983]

• Steve Awodey, Nicola Gambino, Kristina Sojakova, Homotopy-initial algebras in type theory, Journal of the ACM 63 6 (2017) 1–45 [arXiv:1504.05531, doi:10.1145/3006383]

HITs in cubical type theories:

• Thierry Coquand, Simon Huber, Anders Mörtberg, On Higher Inductive Types in Cubical Type Theory, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’18. Oxford, United Kingdom: ACM, 2018, pp. 255–264. [arXiv:1802.01170, doi:10.1145/3209108.3209197]

• Evan Cavallo, Robert Harper, Higher inductive types in cubical computational type theory, Proceedings of the ACM on Programming Languages 3 POPL (2019) 1–27 [doi:10.1145/3290314, pdf]

Further developments:

• Henning Basold, Herman Geuvers, Niels van der Weide, Higher Inductive Types in Programming, Journal of Universal Computer Science 23 1 (2017) 63-88 [pdf, slides pdf]

• Floris van Doorn, §3 in: On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory (2018) [arXiv:1808.10690]

  (formalization in Lean)

• Niccolò Veltri, Niels van der Weide, Constructing Higher Inductive Types as Groupoid Quotients, Logical Methods in Computer Science 17 2 (2021) [arXiv:2002.08150, doi:10.23638/LMCS-17(2:8)2021]

Implementation of HITs in proof assistants:

in Agda:

• Guillaume Brunerie, Implementation of higher inductive types in HoTT-Agda (2016) [github]

  (deprecated)

in Cubical Agda:

• Andrea Vezzosi, Anders Mörtberg, Andreas Abel, §4 in: Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types, Proceedings of the ACM on Programming Languages 3 ICFP 87 (2019) 1–29 [doi:10.1145/3341691, pdf]

• cubical Agda documentation: Higher Inductive Types

in Coq:

• Bruno Barras, Native implementation of Higher Inductive Types (HITs) in Coq PDF

in Lean: van Doorn (2018)

in Narya:

• Higher datatypes and codatatypes, Narya documentation. (web)

Discussion of coinduction via HITs:

• Magnus Baunsgaard Kristensen, Rasmus Ejlers Møgelberg, Andrea Vezzosi, Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks (arXiv:2102.01969)

Discussion of impredicative encodings of higher inductive types:

• Steve Awodey, Jonas Frey, and Sam Speight. “Impredicative Encodings of (Higher) Inductive Types”. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’18. Oxford, United Kingdom: ACM, 2018, pp. 76–85. (ISBN:978-1-4503-5583-4, doi:10.1145/3209108.3209130)