A Categorical Construction for the Computational Definition of Vector Spaces

Abstract

Lambda-\({\mathcal {S}}\) 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-\({\mathcal {S}}\) 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-\({\mathcal {S}}\) 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-\({\mathcal {S}}^{*}\) (a fragment of Lambda-\({\mathcal {S}}\)), 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.