Trace semantics has been defined for various non-deterministic systems with different input/output types, or with different types of "non-determinism" such as classical non-determinism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of "trace semantics" are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in the

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... ory of sets and functions yields a final coalgebra in the Kleisli category, for monads with a suitable order structure. The proof relies on coincidence of limits and colimits, like in the work of Smyth and Plotkin. 1 In this paper we use the terminology non-determinism in its broader sense. It includes: classical non-determinism where one has a set of possible choices; probabilistic non-determinism where one has a probability distribution over possible choices; also systems with non-termination.