A categorical construction for the computational definition of vector spaces
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.
Cite
@article{arxiv.1905.01305,
title = {A categorical construction for the computational definition of vector spaces},
author = {Alejandro Díaz-Caro and Octavio Malherbe},
journal= {arXiv preprint arXiv:1905.01305},
year = {2020}
}
Comments
39 pages. Applied Categorical Structures (2020). arXiv admin note: text overlap with arXiv:1806.09236