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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 . . . . ....
A Taste of Categorical Petri Nets
, 1996
"... This report aims at providing introductory concepts for a categorical approach for the study of Petri Nets. After motivating why a categorical approach for studying petri nets might be desirable, we show that "classical" place/transition Nets, usually seen as bipartite directed graphs, can ..."
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This report aims at providing introductory concepts for a categorical approach for the study of Petri Nets. After motivating why a categorical approach for studying petri nets might be desirable, we show that "classical" place/transition Nets, usually seen as bipartite directed graphs, can be naturally given a monoid structure. Upon this idea we construct a category with place/transition Nets as objects and a suitable notion of place/transition Net morphisms as morphisms. We also verify the existence of ubiquitous categorical constructions in order to verify issues concerning cocompleteness. In the end, the usual notion of semantics of place/transition Nets by a marking graph construction is shown to be nothing else than an adjoint situation between two suitable functors. Further points of interest are pointed out in the bibliographic notes. Contents 1 Introduction 2 2 Preliminaries 3 3 Categorical P/TNets 8 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...