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The paper explores the relation between noncommutative power series andtopological theories of one-dimensional cobordisms decorated by labelledzero-dimensional submanifolds. These topological theories give rise to severaltypes of tensor envelopes of noncommutative recognizable power series,including the categories built from the syntactic algebra and syntactic idealsof the series and the analogue of the Deligne category.

Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series