# A categorical construction for the computational definition of vector spaces

## TL;DR

This paper introduces a categorical semantics for Lambda-S, a lambda calculus extension that models vector spaces and their spans, unifying non-cloning quantum approaches through functor compositions.

## Contribution

It provides an abstract categorical semantics for Lambda-S*, interpreting vector space constructs via adjunctions between Cartesian and monoidal categories.

## Key findings

- Lambda-S models vector spaces using a rewrite system.
- S is interpreted as a composition of functors in an adjunction.
- The semantics reveal the role of a hidden forgetful functor in computational reasoning.

## 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.