Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relativized universal algebra via partial Horn logic

Yuto Kawase

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.

Relativized universal algebra via partial Horn logic

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 . In this paper, using partial Horn theories, we syntactically generalize such an equivalence to arbitrary locally presentable categories from ; 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.
Paper Structure (26 sections, 41 theorems, 69 equations)

This paper contains 26 sections, 41 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Summary
  4. Infinitary relative algebraic theories
  5. Relative algebras
  6. Examples of relative algebraic theories
  7. The finitary case
  8. The infinitary case
  9. Morphisms between relative algebraic theories
  10. Birkhoff's variety theorem
  11. Presentable proper factorization systems
  12. Closed monomorphisms
  13. Birkhoff's variety theorem for partial Horn theories
  14. A syntactic description of accessible monads
  15. From relative algebraic theories to monads
  16. ...and 11 more sections

Key Result

theorem 1

Let $(\Omega, E)$ be a (many-sorted) algebraic theory, where $\Omega,E$ is the sets of operators and equations. Then, a full subcategory $\mathscr{E}\subseteq\mathop{\mathbf{Alg}}(\Omega,E)$ of the category of algebras is definable by equations if and only if it is closed under:

Theorems & Definitions (110)

  • theorem 1: adamek2012birkhoffs
  • remark 1
  • remark 2
  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • definition 2.5
  • remark 3
  • remark 4
  • ...and 100 more