Examples and counterexamples of injective types

2026-01-18

AI Summary

This study investigates the algebraic injectivity of mathematical structures in univalent type theory and its connections to logical principles. By constructing positive examples—such as ordinals, iterative sets, pointed directed-complete partial orders, and ∞-magmas—and negative examples—including the Booleans and Dedekind reals—the paper systematically characterizes the conditions under which types are injective. Key contributions include establishing the injectivity of several common structures, proving that the existence of nontrivial small injective types is equivalent to the weak propositional resizing axiom, and elucidating implications or equivalences between injectivity and principles such as weak excluded middle and propositional projectivity. Furthermore, the work formulates a Rice-type undecidability theorem applicable to such injective types.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

It is known that, in univalent mathematics, type universes, the type of $n$-types in a universe, reflective subuniverses, and the underlying type of any algebra of the lifting monad are all (algebraically) injective. Here, we further show that the type of ordinals, the type of iterative (multi)sets, the underlying type of any pointed directed complete poset, as well as the types of (small) $\infty$-magmas, monoids, and groups are all injective, among other examples. Not all types of mathematical structures are injective in general. For example, the type of inhabited types is injective if and only if all propositions are projective. In contrast, the type of pointed types and the type of non-empty types are always injective. The injectivity of the type of two-element types implies Fourman and \v{S}\v{c}edrov's world's simplest axiom of choice. We also show that there are no nontrivial small injective types unless a weak propositional resizing principle holds. Other counterexamples include the type of booleans, the simple types, the type of Dedekind reals, and the type of conatural numbers, whose injectivity implies weak excluded middle. More generally, any type with an apartness relation and two points apart cannot be injective unless weak excluded middle holds. Finally, we show that injective types have no non-trivial decidable properties, unless weak excluded middle holds, which amounts to a Rice-like theorem for injective types.

Problem

• injective types
• univalent mathematics
• counterexamples
• propositional resizing
• weak excluded middle

Innovation

• injective types
• univalent mathematics
• weak excluded middle
• propositional resizing
• Rice-like theorem