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Elements of Basic Category Theory
, 1996
"... Category theory provides an elegant and powerful means of expressing relationships across a wide area of mathematics. But further than this it has had a considerable impact on the conceptual basis both of mathematics and many parts of theoretical computer science. Important connections in computer s ..."
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Category theory provides an elegant and powerful means of expressing relationships across a wide area of mathematics. But further than this it has had a considerable impact on the conceptual basis both of mathematics and many parts of theoretical computer science. Important connections in computer science include the design of both functional and imperative programming languages, semantic models of programming languages, semantics of concurrency, specification and development of algorithms, type theory and polymorphism, specification languages, algebraic semantics, constructive logic and automata theory. The purpose of this text is to provide a soft stairway to this infectious and attractive field of mathematics. We provide here a careful and detailed explanation of "basic elements", or more precisely, from the elementary definitions to adjoint situations. The general approach used here is to provide a careful motivation for the majority of constructions as well as a detailed presentat...
Category Theory and the SimplyTyped lambdaCalculus
, 1996
"... This report deals with the question on how to provide a categorical model for the simplytyped calculus. We first introduce cartesian closed categories and work in detail a number of results concerning this construction. Next, we present the basic concepts related with the typed calculus, i.e., co ..."
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This report deals with the question on how to provide a categorical model for the simplytyped calculus. We first introduce cartesian closed categories and work in detail a number of results concerning this construction. Next, we present the basic concepts related with the typed calculus, i.e., concrete syntax for terms, occurrence of variables, context substitution and equivalence of terms. Then we present the typing rules and an equational proof system together with reduction rules that model the execution of expressions (programs). The chapter ends with the presentation of a categorical semantics for the calculus and a soundness proof for the equational proof system. The main technical result of this proof is the substitution lemma, which says, basically, that the (operational) concept of substitution can be understood (algebraically) as a composition of two suitable morphisms in a (cartesian closed) category. Contents 1 Cartesian closed categories 2 1.1 Exponentials . . . . ....