The category theoretic understanding of universal algebra: Lawvere theories and monads

Martin Hyland, John Power

Research output: Contribution to journalArticle

50 Citations (Scopus)

Abstract

Lawvere theories and monads have been the two main previous termcategorynext termprevious termtheoreticnext term formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin's mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960's, and how the renewed interest in them in a computer science setting might develop in future.
Original languageEnglish
Pages (from-to)437-458
Number of pages22
JournalElectronic Notes in Theoretical Computer Science
Volume172
DOIs
Publication statusPublished - 1 Apr 2007

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