A categorical construction for the computational definition of vector spaces

Alejandro Díaz-Caro; Octavio Malherbe

doi:10.1007/s10485-020-09598-7

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arXiv · 1905.01305

A categorical construction for the computational definition of vector spaces

Alejandro Díaz-Caro·Octavio Malherbe

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Abstract

Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S* (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.

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BibTeX RIS

Alejandro Díaz-Caro, Octavio Malherbe. 2020-04-11. A categorical construction for the computational definition of vector spaces. https://doi.org/10.1007/s10485-020-09598-7

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