Birkhoff's variety theorem for relative algebraic theories

TL;DR

This paper generalizes the classical correspondence between algebraic theories and finitary monads from sets to locally finitely presentable categories using partial Horn logic, establishing an equivalence with a generalized Birkhoff's variety theorem.

Contribution

It introduces the concept of -relative algebraic theories and proves their equivalence to finitary monads on , extending classical results to broader categorical contexts.

Findings

Defines -relative algebraic theories for locally finitely presentable categories.

Establishes an equivalence between these theories and finitary monads on .

Proves a generalized Birkhoff's variety theorem in this setting.

Abstract

An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathscr{A}$, we define an "algebraic concept" relative to $\mathscr{A}$, which will be called an $\mathscr{A}$-relative algebraic theory, and show that $\mathscr{A}$-relative algebraic theories are equivalent to finitary monads on $\mathscr{A}$. In establishing such equivalence, a generalized Birkhoff's variety theorem plays an important role.