Trace semantics via determinization

This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. In the last few years, two approaches, somewhat incomparable at first sight, captured successfully in a coalgebraic setting trace semantics for various types of transition systems. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories and required some non-trivial assumptions, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). The second development stemmed from the observation that trace semantics can also arise by performing a determinization construction and used final coalgebras in Eilenberg–Moore categories. In this paper, we develop a systematic study in which the two approaches can be studied and compared. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.