Reduction along strong Dirac maps

Abstract We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson momentum maps. We recover a large number of familiar constructions in Poisson and quasi‐Poisson geometry, and we introduce new examples of Poisson, quasi‐Poisson, and Dirac‐reduced structures. In particular, we obtain quasi‐Poisson analogs of several classes of spaces that are studied in geometric representation theory.