Year: 2024
Type: preprint
Abstract: Algebraic theories, sometimes called equational theories, are syntactic notions given by finitary operations and equations, such as monoids, groups, and rings. There is a well-known category-theoretic treatment of them that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, using partial Horn theories, we syntactically generalize such an equivalence to arbitrary locally presentable categories from $\mathbf{Set}$; the corresponding algebraic concepts relative to locally presentable categories are called relative algebraic theories. Finally, we give a framework for universal algebra relative to locally presentable categories by generalizing Birkhoff's variety theorem.
Source: arXiv (Cornell University)
Author: Kawase, Yuto
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4393870910