In physics, a local Lagrangian induces a covariant phase space equipped with a canonical presymplectic form. The quotient of this by symmetries that, in good cases, make the pre-symplectic form a genuine symplectic form, is called the reduced phase space. Generally, given a symplectic manifold or presymplectic manifold or Poisson manifold regarded as a phase space equipped with a suitable (Hamiltonian-) action by a Lie group, the corresponding symplectic reduction or presymplectic reduction or Poisson reduction is, if it exists, the corresponding reduced phase space. Given a local Lagrangian (we display it in codimension 1 mechanics for simplicity of notation) the corresponding covariant phase space is the space of solutions of the Euler-Lagrange equations equipped with the presymplectic form which is exact, with potential as discussed in detail at covariant phase space. Together, this is a prequantum bundle, hence a circle bundle with connection whose curvature is . It so happens that the underlying principal U(1)-bundle of this is trivial, and hence the connection is given by the globally defined differential 1-form . But this trivialility is only superficial: the symmetry group of the Lagrangian is supposed to act by Hamiltonian flows and the prequantum connection is to be equipped with -equivariant connection structure for it to count as a connection on the reduced phase space. Another way to say this, using the higher differential geometry of smooth groupoids: the above prequantum bundle is modulated by a map to the smooth moduli stack of circle bundles with connection, and the -Hamiltonian action induced the action groupoid ; and the construction of the reduced prequantization is the construction of the diagonal morphism in the following diagram (of smooth groupoids) For -dimensional field theory, the local Lagrangian of the above dsicussion arises as the transgression of an -form Lagrangian down in codimension , a refinement discussed in more detail at local...