The coalgebraic µ-calculus is a generic framework for fixpoint logics with varying branching types that subsumes, besides the standard relational µ-calculus, such diverse logics as the graded µ-calculus, the monotone µ-calculus, the probabilistic µ-calculus, and the alternating-time µ-calculus. In the present work, we give a local model checking algorithm for the coalgebraic µ-calculus using a coalgebraic variant of parity games that runs, under mild assumptions on the complexity of the

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... d one-step satisfaction problem, in time p k where p is a polynomial in the formula and model size and where k is the alternation depth of the formula. We show moreover that under the same assumptions, the model checking problem is in NP ∩ coNP, improving the complexity in all mentioned non-relational cases. If one-step satisfaction can be solved by means of small finite games, we moreover obtain standard parity games, ensuring quasi-polynomial run time. This applies in particular to the monotone µ-calculus, the alternating-time µ-calculus, and the graded µ-calculus with grades coded in unary.