Pursuing the notions of ambidexterity and higher semiadditivity as developed by Hopkins and Lurie, we prove that the span $\infty$-category of $m$-finite spaces is the free $m$-semiadditive $\infty$-category generated by a single object. Passing to presentable $\infty$-categories we obtain a description of the free presentable $m$-semiadditive $\infty$-category in terms of a new notion of $m$-commutative monoids, which can be described as spaces in which families of points parameterized by $m$-finite spaces can be coherently summed. Such an abstract summation procedure can be used to give a formal $\infty$-categorical definition of the finite path integral described by Freed, Hopkins, Lurie and Teleman in the context of 1-dimensional topological field theories.

中文翻译：

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双性和有限跨度的普遍性

遵循Hopkins和Lurie提出的矛盾性和半可加性的概念，我们证明了$ m $有限空间的跨度$ \ infty $-类别是由a单个对象。传递到可表示的$ \ infty $-类别，我们获得了关于$ m $-可替换的$ \ infty $-类别的自由可表示的$ m $-象征性$ \ infty $-类别的描述，该概念用$ m $-可交换类mono半群的新概念表示，其中可以将由$ m $个有限空间参数化的点族相干地求和。这样的抽象求和过程可用于给出Freed，Hopkins，Lurie和Teleman在一维拓扑场理论中描述的有限路径积分的形式$ \ infty $-分类定义。