Finiteness spaces, etale groupoids and their convolution algebras

Given a ringR, we extend Ehrhard's linearization process by associating to any pre-finiteness space anR-module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow thisR-module with the convolution product to obtain anR-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 anyhemicompactspace via this construction. As a corollary, any etale 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.