In this paper we introduce the Schutzenberger category $mathbb D(S)$ of a semigroup $S$. It stands in relation to the Karoubi envelope (or Cauchy completion) of $S$ in the same way that Schutzenberger groups do to maximal subgroups and that the local
divisors of Diekert do to the local monoids $eSe$ of $S$ with $ein E(S)$. In particular, the objects of $mathbb D(S)$ are the elements of $S$, two objects of $mathbb D(S)$ are isomorphic if and only if the corresponding semigroup elements are $mathscr D$-equivalent, the endomorphism monoid at $s$ is the local divisor in the sense of Diekert and the automorphism group at $s$ is the Schutzenberger group of the $mathscr H$-class of $S$. This makes transparent many well-known properties of Greens relations. The paper also establishes a number of technical results about the Karoubi envelope and Schutzenberger category that were used by the authors in a companion paper on syntactic invariants of flow equivalence of symbolic dynamical systems.
We prove that the Karoubi envelope of a shift --- defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift --- is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we
show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants --- some new and some old --- are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov-Dyck and Markov-Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property (A) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property (A) is decidable for sofic shifts.
