In my comment on Miles' post which surveyed current terminology I said: "The algebra closure in the weak operator topology then defines the "semigroupoid" C star algbera. Of course this algebra does contain one idempotent for each object, but this is a consequence of taking the algebra- closure of the set of patial isometries defined by the arrows. " The last sentence is incorrect. I have to stop associating "... oid" with invertibility. Carl