Given a ring R , we extend Ehrhard’s linearization process by associating to any pre-finiteness space an R -module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow this R -module with the convolution product to obtain an R -algebra. As examples of pre-finiteness spaces, we study topological spaces with bounded subsets (i.e., included in a compact) taken to be the finitary subsets. We prove that we obtain a finiteness space from any hemicompact space via this construction. As a corollary, any étale Hausdorff groupoid induces a semigroup in pre-finiteness spaces and its associated convolution algebra is complete in the hemicompact case. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.

中文翻译：

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有限空间、étale groupoids 及其卷积代数

给定一个环 R ，我们通过将具有 Lefschetz 拓扑的 R 模关联到任何前有限空间来扩展 Ehrhard 的线性化过程。对于前有限空间范畴中的一个半群，可以赋予这个R-模以卷积乘积，得到一个R-代数。作为前有限空间的例子，我们研究了作为有限子集的有界子集（即包含在一个紧凑型中）的拓扑空间。我们证明我们通过这种构造从任何半紧空间获得了一个有限空间。作为一个推论，任何 étale Hausdorff groupoid 都会在 pre-finiteness 空间中引入一个半群，并且其相关的卷积代数在半紧的情况下是完备的。对于与任何可数行有限有向图相关联的无限路径群尤其如此。