Higher Inductive Types (HITs)

• Higher Inductive Types (HITs) are type-forming constructs that extend ordinary inductive types by including point, path, and higher-path constructors to represent spaces and identifications.
• They embody a homotopy-initial algebra structure, providing recursion and induction principles that enforce computational rules up to homotopy.
• Applications of HITs span modeling spheres, truncations, and quotients, which are pivotal in synthetic homotopy theory, formal topology, and univalent mathematics.

A higher inductive type (HIT) is a type-forming construct that augments ordinary inductive types with path constructors, i.e., constructors that generate identifications (equalities) between elements, and potentially higher-dimensional identifications among these paths. HITs formalize homotopical spaces as types within dependent type theories and provide foundational tools for synthetic homotopy theory, formal topology, and univalent mathematics. They are characterized both by their introduction rules—specifying points, paths, and higher paths—and by universal mapping properties, notably as homotopy-initial algebras.

1. Formal Definition and Presentation

HITs are declared by specifying a finite (possibly infinite) family of constructors:

• Point-constructors: $c_i : \Gamma_i \to X$, introducing elements of the type $X$.
• Path-constructors: $p_j : \Delta_j \to (x_j =_X y_j)$, introducing equalities between point-constructors or previously defined elements of $X$.
• Higher-path-constructors: $$h_k : H_k \to (l_\alpha =_{c_i(x)=c_{i'}(x')} l'_\alpha)$$, introducing identifications between paths.

For example, the circle $S^1$ is presented as

$$\mathsf{base}: S^1 \qquad \mathsf{loop}: \mathsf{base} = \mathsf{base}$$

These presentations generalize ordinary inductive types, enabling the type-theoretic realization of homotopy-theoretic spaces such as spheres, tori, pushouts, and truncations (Sojakova, 2014).

2. Universal Property: Homotopy-Initial Algebra

A central characterization is that a HIT is the homotopy-initial algebra for its signature. For a signature specifying points $C$, paths $B$, arities $A$, and structural maps $f$, $g$, the category of HIT-algebras (Sojakova, 2014) comprises:

• Objects $(X, p, s)$ where $p : C\to X$ and $$s : \prod_{b:B} A(b)\to \mathsf{Id}_X(p f(b), p g(b))$$.
• Morphisms $(h, \beta, \Theta)$ comprising $h : X \to Y$, preserving the algebraic structure and the path identifications.

The HIT itself is defined so that for every other algebra $(Y, q, r)$, the type of algebra homomorphisms from $X$ to $Y$ is contractible. This initiality yields the usual recursion/induction principles up to homotopy.

3. Type-Theoretic Rules: Formation, Introduction, Elimination, Computation

The rules governing HITs are analogous to those for inductive types but include explicit path operations:

• Formation: The type $X$ is introduced.
• Introduction: Point-constructors and path-constructors define elements and identifications, e.g., for $S^1$,
  • $base : S^1$
  • $loop : base = base$
• Elimination (recursion/induction): To define a function $f : X \to Y$, supply images $f(c_i(\gamma))$ and images of paths, with universal properties determined by the initial-algebra semantics.
• Computation: Computation rules are strict for point-constructors, propositional (i.e., up-to-homotopy) for path-constructors due to the intensional nature of identity types.

Example (circle):

• Non-dependent recursor: $\mathrm{rec}_{S^1}(b_0, \ell_0): S^1 \to X$, with
  • $\mathrm{rec}(base) \equiv b_0$
  • $\mathrm{ap(rec)}(loop) = \ell_0$
• Dependent eliminator: $$\mathrm{ind}_{S^1}(P, b_0, \ell_0): \prod_{x:S^1} P(x)$$, with computation $ind(base) \equiv b_0$, $apd(ind)(loop) = \ell_0$ (Doorn, 2018).

4. Key Examples and Constructions

HITs encode a wide variety of spaces and quotients:

• Spheres $S^n$: Presented by base points and $n$-dimensional path constructors (Coquand et al., 2018).
• Suspensions and Pushouts: HITs such as suspensions $\mathrm{Susp}(A)$ and pushouts $A \sqcup_C B$ are specified by points and interval-indexed path constructors, yielding their respective universal properties in homotopy theory (Coquand et al., 2018).
• Truncations $\|A\|_n$: Recursively killing higher paths via universal path constructors to enforce $n$-type structure.
• Set and groupoid quotients: Identifying elements via path constructors, with higher-path constructors to enforce coherence (e.g., groupoid quotient $\mathsf{GQuot}(G)$ (Veltri et al., 2020)).

5. Eliminators, Computation, and Homotopical Semantics

Eliminators for HITs manifest as initiality/induction principles:

• Recursion principle: To define $f : X \to Y$, one assigns $f$ to each constructor, with induced actions on identifications.
• Induction principle: For a family $P : X \to \mathcal{U}$, define $f(x) : P(x)$ respecting the constructors and path constructors (with induced transport for paths).
• Computation rules: Strict (judgmental) on points, and propositional (i.e., via specified paths) for path constructors (Sojakova, 2014, Coquand et al., 2018, Kraus et al., 2019).

Semantically, HITs are modeled as initial algebras for polynomial endofunctors in suitably structured $(\infty,1)$-categories or model categories (Lumsdaine et al., 2017, Uemura, 2024). Cell-monad constructions structure the semantics analogue to CW-complex attachments.

6. Advanced Variants: QIITs and HIITs

Extensions include:

• Quotient Inductive-Inductive Types (QIITs): Simultaneous definitions of multiple types with mutual dependencies and path constructors (Altenkirch et al., 2016).
• Higher Inductive-Inductive Types (HIITs): Generalizing HITs to simultaneous definitions of several sorts (types), each with point and path constructors, indexed over each other (Kaposi et al., 2019).

Induction principles for these are derived via signatures and categorical constructions generalizing those for HITs.

7. Implementation and Formalization in Proof Assistants

Practical realization of HITs in type-theory-based proof assistants involves:

• Licata's Private Inductive Technique: Postulating point-constructors privately, attaching path-constructors as axioms, and defining eliminators inside a scoped module for definitional computation (Bauer et al., 2016).
• Metaprogramming and Code Generation: Agda's elaborator reflection automates the boilerplate for both recursors and inductors, with correct computation rules for point and path constructors (Vivekanandan, 2018).

Libraries such as the HoTT Library (Coq), Lean's homotopy type theory modules, and Agda reflect the spectrum of currently available methods for HITs.

8. Applications and Impact

HITs provide foundations for:

• Formal topology and the representation of spaces up to homotopy (e.g., spheres, tori, Postnikov towers) (Doorn, 2018).
• Quotients and truncations critical for set-level mathematics in univalent foundations (Doorn, 2015).
• Synthetic homotopy theory, spectral sequence calculations, and categorical constructions crucial in modern mathematics.

The universal property and induction principle of HITs enable concise reasoning about spaces and identifications within type-theoretic provenance.

9. Open Questions and Ongoing Research

Current research addresses:

• Extension and semantic modeling of recursive and higher-dimensional HITs (notably, extending beyond 1-paths or finite constructors) (Lumsdaine et al., 2017, Altenkirch et al., 2020).
• The categorical semantics of HITs in $(\infty,1)$-categories and set theory without choice, including limitations of constructive realizability (Swan, 2020, Uemura, 2024).
• The interplay of computational canonicity, definitional equalities, and the formulation of conditions/coconditions for path and higher-path computation (Zhang et al., 2024).

The precise boundaries of inductive type formation, computation up to higher path, and categorical presentability continue to be the subject of theoretical and mechanized inquiry.

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References:

• "Higher Inductive Types as Homotopy-Initial Algebras" (Sojakova, 2014)
• "On Higher Inductive Types in Cubical Type Theory" (Coquand et al., 2018)
• "Semantics of higher inductive types" (Lumsdaine et al., 2017)
• "Constructing Higher Inductive Types as Groupoid Quotients" (Veltri et al., 2020)
• "Code Generation for Higher Inductive Types" (Vivekanandan, 2018)
• "The HoTT Library: A formalization of homotopy type theory in Coq" (Bauer et al., 2016)
• "Quotient inductive-inductive types" (Altenkirch et al., 2016)
• "Signatures and Induction Principles for Higher Inductive-Inductive Types" (Kaposi et al., 2019)
• "Path Spaces of Higher Inductive Types in Homotopy Type Theory" (Kraus et al., 2019)
• "A class of higher inductive types in Zermelo-Fraenkel set theory" (Swan, 2020)
• "Impredicative Encodings of (Higher) Inductive Types" (Awodey et al., 2018)
• "Constructing the Propositional Truncation using Non-recursive HITs" (Doorn, 2015)
• "Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks" (Kristensen et al., 2021)
• "(Co)condition hits the Path" (Zhang et al., 2024)
• "The Integers as a Higher Inductive Type" (Altenkirch et al., 2020)
• "Higher inductive types in $(\infty,1)$-categories" (Uemura, 2024)