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 $\mathbf{Set}$. 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.