The Wehrheim-Woodward category of linear canonical relations between \(G\)-spaces (Q6657478)

The problem of making canonical relations between symplectic manifolds into the morphisms of a category was settled in a seminal paper by \textit{K. Wehrheim} and \textit{C. T. Woodward} [Quantum Topol. 1, No. 2, 129--170 (2010; Zbl 1206.53088)], an abstract version of which was given by \textit{D. Li-Bland} and the author [SIGMA, Symmetry Integrability Geom. Methods Appl. 10, Paper 100, 31 p. (2014; Zbl 1325.53116)], showing that the morphisms in the WW-category of linear canonical relations between finite-dimensional symplectic vector spaces could be identified with the pair \(\left( L,k\right) \) consisting of a linear canonical relation and a nonnegative integer, called indexed canonical relations.\N\NThis paper aims to extend the result to the situation where a compact group \(G\) is acting on the linear symplectic spaces, and the linear canonical relations are equivalent, in which the nonnegative integer \(k\) is replaced by an arbitrary isomorphism class \(\mathcal{E}\) of finite-dimensional \(G\)-spaces without symplectic structure involved. The author extends to all highly selective categories \(\boldsymbol{C}\) the main result of [\textit{A. Weinstein}, J. Geom. Mech. 3, No. 4, 507--515 (2011; Zbl 1257.53111)] to the effect, in the special case where there is a certain highly selective category of relations between symplectic manifolds, with some morphisms the smooth canonical relations, every morphism \(Y\rightarrow X\) in \(\mathrm{WW}\left( \boldsymbol{C}\right) \) may be represented by a composition \(Y\rightarrowtail Q\twoheadrightarrow X\) of just two suave morphisms in \(\boldsymbol{C}\), where the decorations on the arrows mean that \(Q\twoheadrightarrow X\) is a reduction and \(Y\rightarrowtail Q\) is a coreduction.