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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 . . . . ....