As shown by Tsukada and Ong, normal (extensional) simply-typed resource terms correspond to plays in Hyland-Ong games, quotiented by Melli\`es' homotopy equivalence. Though inspiring, their proof 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：正如冢田和Ong所示，正规（外延的）简单类型化资源项对应于Hyland-Ong博弈中的对局，商掉Melliès的同伦等价。尽管这一结果富有启发性，但其证明是间接的，依赖于关系模型在对应关系双方的单射性——具体而言，资源演算的动态性仅通过关系模型与由正规化定义的正规项合成的兼容性来体现。本文中，我们重新审视并扩展了这些结果。第一项贡献是重新表述对应关系，引入我们称为“增广”的因果结构，它们是Hyland-Ong对局在同伦意义下的典范代表。这使我们可以直接且显式地阐述与正规资源项的联系。作为第二项贡献，我们将这一阐述扩展到资源项的归约：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一个指称模型，该模型在归约下保持不变。关键步骤——也是我们的第三项贡献——是提出一个称为资源范畴的范畴模型，它对资源演算的意义正如微分范畴对微分λ-演算的意义。