As shown by Tsukada and Ong, simply-typed, normal and eta-long resource terms correspond to plays in Hyland-Ong games, quotiented by Melli\`es' homotopy equivalence. The original proof of this inspiring result is indirect, relying on the injectivity of the relational model w.r.t. both sides of the correspondence -- in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization. In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step -- and our third contribution -- is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential lambda-calculus.

翻译：正如塚田与Ong所示，简单类型化、范式且η-长的资源项对应于Hyland-Ong博弈中经Melliès同伦等价商化的对局。这一启发式结论的原始证明是间接的，依赖关系模型在对应关系两侧的单射性——具体而言，资源演算的动态性仅通过关系模型与由规范化定义的范式项组合的兼容性被纳入考量。本文重新审视并扩展了这些结果。我们的第一个贡献是：通过引入称为"增广"的因果结构（即Hyland-Ong博弈对局在忽略同伦意义下的典范代表），重新表述该对应关系，从而直接且明确地阐明了与范式资源项的联系。作为第二个贡献，我们将这一阐释扩展至资源项的归约：基于将策略视为增广的加权和这一概念，我们构建了资源演算的指称模型，该模型在归约下具有不变性。关键步骤——也是我们的第三个贡献——是提出称为"资源范畴"的范畴模型，它对资源演算的作用恰如微分范畴对微分λ-演算的作用。