Polynomial Universes in Homotopy Type Theory

Awodey, later with Newstead, showed how polynomial functors with extra structure (termed ``natural models’’) hold within them the categorical semantics for dependent type theory. Their work presented these ideas clearly but ultimately led them outside of the usual category of polynomial functors to a particular \emph{tricategory} of polynomials in order to explain all of the structure possessed by such models. This paper builds off that work – explicating the categorical semantics of dependent type theory by axiomatizing them entirely in terms of the usual category of polynomial functors. In order to handle the higher-categorical coherences required for such an explanation, we work with polynomial functors in the language of Homotopy Type Theory (HoTT), which allows for higher-dimensional structures to be expressed purely within this category. The move to HoTT moreover enables us to express a key additional condition on polynomial functors – \emph{univalence} – which is sufficient to guarantee that models of type theory expressed as univalent polynomials satisfy all higher coherences of their corresponding algebraic structures, purely in virtue of being closed under the usual constructors of dependent type theory. We call polynomial functors satisfying this condition \emph{polynomial universes}. As an example of the simplification to the theory of natural models this enables, we highlight the fact that a polynomial universe being closed under dependent product types implies the existence of a distributive law of monads, which witnesses the usual distributivity of dependent products over dependent sums.

💡 Research Summary

The paper revisits the relationship between polynomial functors and the categorical semantics of dependent type theory, originally explored by Awodey and later extended with Newstead through the notion of “natural models”. While natural models elegantly solve the strictness problem—ensuring that substitution and type‑forming operations satisfy Beck‑Chevalley conditions strictly—they require a sophisticated tricategory of polynomial functors to accommodate the higher‑dimensional coherence data. This tricategorical machinery is both conceptually heavy and technically cumbersome, especially for formalisation.

The authors propose a different route: internalising polynomial functors directly in Homotopy Type Theory (HoTT). By working in HoTT, all higher‑dimensional equalities are represented as paths, and the univalence axiom guarantees that equivalences between types can be identified with equalities. Consequently, a polynomial functor that is univalent automatically carries the necessary higher coherence, eliminating the need for an external tricategory.

A polynomial functor is defined as a pair ((A, B)) where (A) is an index type and (B : A \to \mathsf{Type}) assigns a fibre type to each index. In Agda this is expressed as a Σ‑type over a universe level. Morphisms between polynomial functors are given by lenses: a forward map (f : A \to C) together with a family of backward maps (g_a : D(f(a)) \to B(a)). When each (g_a) is an equivalence, the lens is called Cartesian. A Cartesian lens from a polynomial (p) to a polynomial universe (u) precisely encodes that (u) is closed under the type former represented by (p).

The paper shows that the usual type formers Σ and Π can be represented by specific polynomials. Existence of a Cartesian lens (u \otimes u \to u) (where (\otimes) is the monoidal product of polynomials given by composition) yields closure under Σ‑types; the lens’s forward component builds the Σ‑type, while its backward component provides the elimination rules and an equivalence witnessing the β‑η laws. Closure under Π‑types is captured via a vertical‑Cartesian factorisation and a distributive law of monads: a map (δ : u \otimes u \to u) satisfying the usual monad distributivity equations gives the Π‑type formation and its interaction with Σ‑types. In this way the familiar distributivity of dependent products over dependent sums emerges automatically from the categorical structure of polynomial universes.

All the main constructions—definition of the polynomial category, Cartesian lenses, the monoidal product, the Σ‑ and Π‑closure conditions, and the distributive law—are formalised in Agda. The mechanised proofs confirm that the HoTT‑based approach faithfully reproduces the required coherence without external higher‑categorical scaffolding.

The contributions are twofold. First, they replace the external tricategory with internal HoTT reasoning, dramatically simplifying the meta‑theory while preserving full higher‑dimensional coherence via univalence. Second, they identify polynomial universes as robust models of dependent type theory that support all basic type formers and enjoy a rich monadic structure, including a canonical distributive law. This opens the door to a new “polynomial monad” perspective on type theory, with potential applications to ∞‑category‑based logic, homotopical programming languages, and the formalisation of mathematics.

In summary, by axiomatizing dependent type theory entirely within the ordinary category of polynomial functors—interpreted inside HoTT—the authors provide a cleaner, more tractable foundation for natural models, demonstrate how Σ‑ and Π‑type closure follows from Cartesian lenses, and reveal a built‑in distributive law of monads that captures the familiar interaction of dependent products and sums. The work both clarifies the categorical semantics of type theory and supplies a concrete, machine‑checked implementation, suggesting a promising avenue for future research in higher‑dimensional type theory and its applications.