Abstract
This paper forms part of a project focusing on the development of homological and simplicial methods in rewriting. The purpose of this contribution is to generalize Kobayashi's theorem for monoids to “monoids with several objects”. Following Squier's theorem, Kobayashi constructs a resolution for monoids presented by convergent rewriting systems. We construct a free acyclic resolution for kC, as a C-bimodule over a commutative ring k, where C is a small category provided with a convergent presentation. This resolution, associated to the Knuth-Bendix completion algorithm, reflects the combinatorial properties of C. In particular, categories admitting finite convergent presentations by graphs and relations have a finite type Hochschild-Mitchell homology.
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