TY - JOUR

T1 - The category theoretic understanding of universal algebra: Lawvere theories and monads

AU - Hyland, Martin

AU - Power, John

PY - 2007/4/1

Y1 - 2007/4/1

N2 - 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.

AB - 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.

UR - http://dx.doi.org/10.1016/j.entcs.2007.02.019

U2 - 10.1016/j.entcs.2007.02.019

DO - 10.1016/j.entcs.2007.02.019

M3 - Article

VL - 172

SP - 437

EP - 458

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

ER -