# Decomposing typed lambda calculus into a couple of categorical programming languages

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## Abstract

We give two *categorical programming languages* with *variable arrows* and associated abstraction/reduction mechanisms, which extend the possibility of *categorical programming* [Hag87, CF92] in practice. These languages are complementary to each other — one of them provides a first-order programming style whereas the other does higher-order — and are “children” of the *simply typed lambda calculus* in the sense that we can decompose typed lambda calculus into them and, conversely, the combination of them is equivalent to typed lambda calculus. This decomposition is a consequence of a semantic analysis on typed lambda calculus due to C. Hermida and B. Jacobs [HJ94].

## Keywords

Reduction Rule Left Adjoint Lambda Calculus Functional Completeness Inclusion Functor
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