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A concrete model for a typed linear algebraic lambda calculus
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- Journal:
- Mathematical Structures in Computer Science / Volume 34 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 21 November 2023, pp. 1-44
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We give an adequate, concrete, categorical-based model for Lambda-
${\mathcal S}$, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-
${\mathcal S}$ is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda-
${\mathcal S}$ has a superposition constructor S such that a type A is considered as the base of a vector space, while SA is its span. Our model considers S as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over 
$\mathbb C$. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
