In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding $\infty$ -categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted $K$ -theory.

中文翻译：

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回缩空间中通过对称谱的乘法参数化同伦理论

为了处理扭曲（共）同调中的乘法现象，我们为参数化同伦理论引入了一个新的点集级框架。我们提供了一个卷积粉碎产品，它下降到相应的 $\infty$ -分类产品，并允许方便地构造交换参数化环谱。作为一个直接应用，我们比较了广义 Thom 谱的各种模型。在一篇配套论文中，这种方法用于比较扭曲的同伦和算子代数模型 $K$ -理论。