Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as uniform approach to taking quotients in geometry. This construction holds categories smooth manifolds, complex analytic spaces, and algebraic varieties, has an interpretation terms derived stacks shifted It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein pre-images Poisson transversals under moment maps, cutting, implosion, Ginzburg–Kazhdan Moore–Tachikawa varieties topological quantum field theory. A key feature our is concrete systematic association Hamiltonian $G$ -space $\mathfrak {M}_{G, S}$ each pair $(G,S)$ , where any Lie group $S\subseteq \mathrm {Lie}(G)^{*}$ satisfying certain non-degeneracy conditions. The spaces satisfy universal property for which generalizes that imploded cross-section. Although these -spaces are explicit natural from Lie-theoretic perspective, some them appear be new.