[Paper Review] Constructing Initial Algebras Using Inflationary Iteration

Andrew M. Pitts, S. C. Steenkamp|arXiv (Cornell University)|Jan 1, 2021

Logic, Reasoning, and Knowledge30 references1 citations

TL;DR

This paper presents a constructive proof of initial algebra existence for endofunctors using inflationary iteration over a notion of 'size' that abstracts from ordinals, avoiding classical logic and the Axiom of Choice. It establishes that, under the weak choice principle WISC, this method applies to a broad class of functors—including those involving quotients and infinite exponentials—enabling initial algebra construction in constructive settings like toposes.

ABSTRACT

An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.

Motivation & Objective

• To develop a constructive alternative to Adámek’s classical theorem on initial algebras, which relies on transfinite iteration over ordinals and classical logic.
• To address the challenge of constructing initial algebras in constructive settings, particularly for functors involving quotients and infinite exponentials, where classical choice principles are problematic.
• To show that initial algebras can be constructed using a notion of 'size' that generalizes ordinals by focusing only on transitive, directed, and well-founded properties, enabling constructive reasoning.
• To demonstrate that the Weakly Initial Sets of Covers (WISC) axiom suffices to ensure the existence of sufficiently large sizes for colimit construction, making the method applicable to a rich class of functors in topos-theoretic models.
• To provide a foundation for modeling dependent type theories with inductive and quotient constructions in constructive toposes, where classical logic does not hold.

Proposed method

• Introduce a new notion of 'size' as a well-founded, transitive, and directed order, abstracting from limit ordinals while preserving the essential properties needed for colimit construction.
• Define inflationary iteration of an endofunctor F over a size κ, constructing a diagram (Fα0)α∈κ in a cocomplete category C, where each Fα0 is the image of F applied iteratively up to α.
• Use the universal property of colimits to define linking morphisms iα: Fα0 → Fα+0, ensuring compatibility across the diagram.
• Prove that if the colimit of this inflationary iteration exists and the final linking morphism iα is an isomorphism, then (Fα0, iα⁻¹) forms an initial F-algebra.
• Establish that sizes exist with upper bounds for any given signature of indexes, using constructive principles inspired by Taylor’s treatment of ordinals.
• Apply the WISC axiom to ensure that for any functor F, there exists a size κ such that F preserves colimits of shape κ, enabling the construction of initial algebras without classical choice.

Experimental results

Research questions

• RQ1 Can initial algebras for endofunctors be constructed in a constructive logic setting without relying on the Axiom of Choice or the Law of Excluded Middle?
• RQ2 Can the classical transfinite iteration over ordinals be replaced by a more general notion of 'size' that retains the necessary properties for colimit formation while being amenable to constructive reasoning?
• RQ3 What choice principle is sufficient to ensure the existence of sufficiently large sizes for colimit construction in constructive category theory?
• RQ4 To what extent can this constructive method be applied to non-polynomial functors involving infinite exponentials and quotient constructions?
• RQ5 Can this approach be extended to provide constructive proofs of existence for free algebras in more general equational systems, particularly in toposes?

Key findings

• A new constructive theorem (Theorem 3.8) is established, showing that if an endofunctor F preserves colimits of shape κ for some size κ, then the colimit of its inflationary iteration over κ yields an initial F-algebra.
• The notion of 'size' used in the construction generalizes ordinals by focusing on transitive, directed, and well-founded order structures, avoiding reliance on classical properties of ordinals.
• The existence of sizes with upper bounds for any given signature of indexes is proven constructively, ensuring that the method applies broadly.
• The Weakly Initial Sets of Covers (WISC) axiom is shown to be sufficient to guarantee the existence of such sizes for a rich class of functors, including those involving quotients and infinite exponentials.
• The method avoids the use of a stationary size ∞, which can lead to logical inconsistencies in type theories like Agda’s sized types, by instead requiring a colimit construction to obtain the initial algebra.
• The approach is predicative and valid in elementary toposes with natural number objects and universes, making it suitable for modeling dependent type theories with inductive and quotient constructions.