Lawvere theories and monads have been the two main category theoretic 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.
Hyland and Power REF reviewed the history of both Lawvere theories and monads as an approach to universal algebra, how monads came to the fore, Plotkin's work, and the renewed interest in Lawvere theories in a computer science setting.
The category theoretic understanding of universal algebra: Lawvere theories and monads
Elect. Notes in Theor. Comp. Sci., 172:437–458
Computer Science