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

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