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The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... 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 ..."
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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.
Synthetic Differential Geometry: A Way to Intuitionistic Models of General Relativity in Toposes
, 1996
"... W.Lawvere in [4] suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all setti ..."
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W.Lawvere in [4] suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E , particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock [1] in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a some nondegenerate commutative ring R of a line type in E . In this work we shall show that SDG allows to develop a Riemannian geometry and write the Einstein's equations of a field on pseudoRiemannian formal manifold. This give a way for constructing a intuitionistic models of general relativity in suitable toposes. 1 Preliminaries In this paper will be given some metrical notions i...