The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $\infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.

中文翻译：

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卷积乘积的同伦不变性

本文的目的是表明各种卷积乘积是完全同伦的，这意味着它们在没有任何共纤假设的情况下在两个变量中保持弱等价。我们为有限集和注入类别索引的单纯集图以及驯服的 $M$-单纯集建立了这个属性，其中 $M$ 是正自然数的单射自映射的幺半群。我们还表明，Nikolaus 和第一作者研究的某个卷积产品是完全同伦的。这意味着每个对称的幺半群$\infty$-category 都可以用一个完全同伦幺正积的对称幺半群模型类别来表示。