Aspects of Predicative Algebraic Set Theory I: Exact Completion

This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability and the other on Sheaf Models in Algebraic Set Theory.

💡 Research Summary

This paper inaugurates a three‑part series on Predicative Algebraic Set Theory (PAST), laying the categorical foundations required for the subsequent treatments of realizability models and sheaf models. The central object of study is a regular category equipped with a class of “small maps” – morphisms that play the role of set‑theoretic functions and satisfy stability under pullback, closure under composition, and a suitable notion of monomorphism. These small maps enable the internal construction of basic set‑theoretic operations such as pairing, function spaces, and power objects, while preserving the predicative nature of the theory.

The main technical contribution of the first part is a thorough analysis of the exact completion process. Given a regular category 𝒞 with a small‑map class S, the exact completion 𝒞̂ is obtained by freely adding effective quotients for all equivalence relations (i.e., by formally adjoining coequalisers of kernel pairs). The authors prove a “small‑map preservation theorem”: the class S lifts to a class Ŝ of small maps in 𝒞̂, and Ŝ retains all the defining properties of small maps. Consequently, the exact completion does not destroy the ability to interpret the basic set‑theoretic constructions that depend on small maps.

To achieve this result, the paper develops a detailed machinery for handling relational objects, kernel pairs, and effective equivalence relations inside 𝒞. It shows that any small map in 𝒞 yields a small relational object whose quotient in the exact completion remains small. The proof hinges on constructing explicit factorisations of morphisms into a small map followed by a regular epimorphism, and then demonstrating that the regular epimorphism becomes an effective quotient after completion.

Beyond preservation, the authors address the existence of a “universe” in the completed category. They formulate sufficient conditions under which 𝒞̂ possesses a universal small map that classifies all other small maps, together with exponentials and power objects for this universe. This universe axiom is crucial for interpreting higher‑order set theory and for ensuring that the exact completion is a genuine model of PAST rather than a merely technical extension.

The paper concludes by outlining how these results feed into the next two installments. In the realizability part, the exact completion will be applied to categories of assemblies and modest sets, providing a predicative setting for Kleene’s realizability and for extracting computational content from set‑theoretic statements. In the sheaf part, the exact completion supplies the necessary exactness to define sheaf models over sites equipped with a small‑map structure, thereby enabling the construction of Grothendieck‑type universes in a predicative context.

Overall, the article establishes that the exact completion is a robust categorical operation that preserves the delicate small‑map structure essential to predicative algebraic set theory, and it prepares the ground for sophisticated model‑theoretic applications in both realizability and sheaf‑theoretic frameworks.