![]() ![]() Here HOM denotes the usual set of morphisms from one object to another, while HOM ( x × a, b ) ≅ HOM ( a, hom ( x, b ) ) HOM(x \times a, b) \cong HOM(a,hom(x,b)) In other words, we have a natural isomorphism These are categories with finite products such that the operation He showed that just as algebraic theoriesĬan be regarded as certain special categories, so can theoriesįormulated in the lambda calculus: to be precise, these correspond toĬartesian closed categories or CCCs for short. Curry: Essays on Combinatory Logic, Lambda CalculusĪnd Formalism, eds. Joachim Lambek, From lambda calculus to Cartesian closed categories, It’s used at over 100 colleges and universities, and it has spawned a secret society called The Knights of the Lambda Calculus, whose emblem celebrates the ability of the λ-calculus to do recursion: It cites pioneers like Haskell Curry, and it even has a big λ on the cover!įans of Oz and the Wizard will be pleased to hear that students call it “the wizard book”, for the obvious reason. Sussman, Structure and Interpretation of Computer Programs. Introductory course in computer science, which is available online: Of the lambda calculus is evident in the textbook developed for MIT’s Its Syntax and Semantics, North-Holland, 1984. This began a long and fruitful line of research - see for example this: Landin, A correspondence between ALGOL 60 and Church’s Lambda calculus can actually be seen as proofs. Is also computable by a Turing machine, and according to theĬhurch-Turing thesis these are all the functions computable by any Any function computable by the lambda calculus Lambda calculus, invented by Church and Kleene in the 1930s as a Treating proofs as computations may seem strained, but itīecomes less so when we move to richer formalisms which allow for moreĬomplex logical reasoning. Is a set of assumptions, while the “output” is the equation to be Proof in Th ( Grp ) Th(Grp), or indeed in any algebraic theory, can be seenĪs a rudimentary form of computation. The relevance of all this to computer science becomes visible when we note that a Model was a functor - hence his thesis title, “Functorial Semantics”. In fact, universal algebra was around long before Lawvere introducedĪlgebraic theories he just modernized it with the realization that a In the case of algebraic theories, the syntax often goes by For example, starting from Th ( Grp ) Th(Grp) weĬan prove consequences of the group axioms merely by jugglingĮquations. The basic idea is simple: if for example T = Th ( Grp ) T = Th(Grp), then Z Z maps the abstract concept of “group” to a specific set, the abstract concept of “multiplication” to a specific multiplication on the chosen set, and so on, thusĭual to the concept of semantics is the concept of syntax, whichĭeals with symbol manipulation. Technically, an algebraic theory T T is a category withįinite products, and a model is a functor that preserves finite products:įrom T T to the category of sets. For example, a “model” of Th ( Grp ) Th(Grp) Loosely speaking, a model is just one of the things The role of semantics enters when we consider “models” of anĪlgebraic theory. (multiplication, inverses…) satisfying equations, this concept canīe formalized using an algebraic theory called Th ( Grp ) Th(Grp). ForĮxample, since the concept of a “group” involves only some operations Categorical semantics was born in Lawvere’s celebrated 1963 thesis on algebraic theories:įunctorial Semantics of Algebraic Theories.įormalism for reasoning about operations that satisfy equations.
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