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.

In this paper, we continue with the algebraic study of Krivine’s realizability, completing and generalizing some of the authors’ previous constructions by introducing two categories with objects the abstract Krivine structures and the implicative algebras, respectively. These categories are related by an adjunction whose existence clarifies many aspects of the theory previously established. We also revisit, reinterpret, and generalize in categorical terms, some of the results of our previous work such as: the bullet construction, the equivalence of Krivine’s, Streicher’s, and bullet triposes and also the fact that these triposes can be obtained – up to equivalence – from implicative algebras or implicative ordered combinatory algebras.

In this work, we continue our consideration of the constructions presented in the paper Krivine's Classical Realizability from a Categorical Perspective by Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented, mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.

We propose the new concept of Krivine ordered combinatory algebra ( $\mathcal{^KOCA}$ ) as foundation for the categorical study of Krivine's classical realizability, as initiated by Streicher (2013).

We show that $\mathcal{^KOCA}$ 's are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. The difference between the two representations is that the elements of a $\mathcal{^KOCA}$ play both the role of truth values and realizers, whereas truth values are sets of realizers in $\mathcal{AKS}$ s.

To conclude, we give a direct presentation of the realizability interpretation of a higher order language in a $\mathcal{^KOCA}$ , which showcases the dual role that is played by the elements of the $\mathcal{^KOCA}$ .