Core Concepts

Formalizing algebras and equations in the UniMath framework for universal algebra.

Abstract

The article presents a library for Universal Algebra in UniMath, focusing on multi-sorted signatures, term algebras, and equation systems. It introduces term algebras without general inductive constructions, enriching UniMath with homotopy W-types. The formalization includes constructions of univalent categories of algebras and equational algebras using displayed categories. The computational relevance is showcased through examples from algebra and propositional logic.

Stats

"UniMath" framework dealing with multi-sorted signatures.
Single sorted ground term algebras are instances of homotopy W-types.
Term algebra over a signature is the initial object of the category of algebras.

Quotes

"We present an implementation of the basics of universal algebra in univalent foundations within the formal environment of UniMath."
"UniMath provides a minimalist implementation of univalent type theory."
"Our formalisation introduces central notions concerning multi-sorted signatures."

Key Insights Distilled From

by Gianluca Ama... at **arxiv.org** 03-18-2024

Deeper Inquiries

The implementation in UniMath enhances the computational nature by providing a framework for formalizing universal algebra within univalent foundations. By defining algebras, equations, and term structures computationally, the library allows for the manipulation and evaluation of algebraic structures using proof assistants. The use of reverse Polish notation and value stacks to represent terms ensures that computations can be carried out efficiently within the system. Additionally, by proving properties such as induction principles on terms and ensuring that operations are evaluable without errors or stack underflows, UniMath enables practical reasoning about algebraic systems.

Introducing displayed categories in constructing univalent categories has significant implications for organizing and reasoning about algebraic structures. Displayed categories provide a way to define structured relationships between objects in a category over another base category. In the context of universal algebra, this allows for modular construction of categories of algebras and equational algebras based on sorted hSets. By utilizing displayed categories, it becomes easier to prove properties like univalence between isomorphic objects in these constructed categories while maintaining coherence with higher-dimensional categorical reasoning.

The concept of homotopy W-types plays a crucial role in formalizing term algebras within UniMath's framework. By structuring ground term algebras as instances of homotopy W-types, it enriches the computational behavior of these term structures. This approach provides a robust foundation for defining recursive data types like terms over signatures without resorting to general inductive constructions not allowed in UniMath. The homotopy W-type structure ensures that term algebras have well-defined recursion principles and induction rules that align with constructive mathematics principles while retaining their computational nature.

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