2023/04/10 by Yuto Kawase, Kawase, Yuto
Computer Science · Mathematics · #18C10 #18C15 #18C35 #18E45 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2304.04382
openalex publication_date 2023/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Set. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category \mathscrA, we define an "algebraic concept" relative to \mathscrA, which will be called an \mathscrA-relative algebraic theory, and show that \mathscrA-relative algebraic theories are equivalent to finitary monads on \mathscrA. In establishing such equivalence, a generalized Birkhoff's variety theorem plays an important role.