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：如Tsukada和Ong所示，规范（外延的）简单类型资源项对应于Hyland-Ong博弈中的对局，并模去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两边的单射性——特别是，资源演算的动态性仅通过关系模型与由规范化定义的规范项复合的兼容性被间接考虑。在本文中，我们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来重新表述该对应关系，这些结构是Hyland-Ong博弈中模去同伦的规范代表元。这使我们能够直接而显式地建立与规范资源项的联系。作为第二个贡献，我们将这一刻画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，我们给出了资源演算的一个在归约下不变的外延模型。一个关键步骤——也是我们的第三个贡献——是称为资源范畴的范畴模型，它对资源演算的作用正如微分范畴对微分λ-演算的作用。