The symmetry enriched center functor is fully faithful
It was shown in [\textit{L. Kong} and \textit{H. Zheng}, Adv. Math. 339, 749--779 (2018; Zbl 1419.18013)] that the Drinfeld center can be made functorial and fully faithful provided that proper domain and codomain are chosen. This Drinfeld center provides a precise and complete mathematical description of the boundary-bulk relation of 2+1D anomaly-free topological orders with gapped boundaries [\textit{L. Kong} et al., Nucl. Phys., B 922, 62--76 (2017; Zbl 1373.82093)]. In the theory of topological orders, topological orders with symmetry is called a symmetry enriched topological (SET) order [\textit{M. Levin} and \textit{Z.-C. Gu}, ``Braiding statistics approach to symmetry protected topological phases'', Phys. Rev. B 86, No. 11, Article ID 115109, 15 p. (2012; \url{doi:10.1103/PhysRevB.86.115109}); \textit{X. Chen} et al., ``Symmetry protected topological orders and the group cohomology of their symmetry group'', Phys. Rev. B 87, No. 15, Article ID 155114, 48 p. (2013; \url{doi:10.1103/PhysRevB.87.155114}); \textit{M. Barkeshli} et al., ``Symmetry fractionalization, defects, and gauging of topological phases'', Phys. Rev. B 100, No. 11, Article ID 115147, 99 p. (2019; \url{doi:10.1103/PhysRevB.100.115147})]. This paper gives a generalization of the work in [\textit{L. Kong} and \textit{H. Zheng}, Adv. Math. 339, 749--779 (2018; Zbl 1419.18013)] to describe the symmetry enriched case, more precisely, to find categorical description of SET orders and give a notion of the symmetry enriched center and make it functorial by choosing proper domain and codomain categories. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] gives preliminaries on finite monoidal categories. \item[\S 3] introduces a series of symmetry enriched categories, especially including finite monoidal categories over \(\mathcal{E}\) and finite braided monoidal categories containing \(\mathcal{E}\), and defines a new tensor product between the latter, named the relative tensor product over \(\mathcal{E}\), via considering a structure named \(\mathcal{E}\)-module braidings, and proves some of their properties. \S 3.7 is devoted to giving another description of relative tensor product over \(\mathcal{E}\) and establishing that it coincides with the one defined in [\textit{T. Lan} et al., Commun. Math. Phys. 351, No. 2, 709--739 (2017; Zbl 1361.81184)], where it is shown that the symmetry can be characterized by a symmetric fusion category \(\mathcal{E}\) over a field \(k\), and it is also shown that one can use a unitary fusion category over \(\mathcal{E}\) to describe the excitations of a 1+1D SET order and a triple \(\left( \mathcal{E},\mathcal{C},\mathcal{M}\right) \) with a unitary modular tensor category \(\mathcal{C}\) over \(\mathcal{E}\) and a minimal modular extension \(\mathcal{M}\) of \(\mathcal{C}\) [\textit{C. F. Venegas-Ramírez}, ``Minimal modular extensions for super-Tannakian categories'', Preprint, \url{arXiv:1908.07487}; \textit{C. Galindo} and \textit{C. F. Venegas-Ramírez}, ``Categorical Fermionic actions and minimal modular extensions'', Preprint, \url{arXiv:1712.07097}; \textit{M. Müger}, Adv. Math. 150, No. 2, 151--201 (2000; Zbl 0945.18006)] to characterize a 2+1D anomaly-free SET order. \item[\S 4] constructs domain and target caterories with symmetry enriched in \(\mathcal{E}\), and equips them with proper tensor products, and finally makes the notion of the center functorial and fully faithful. \item[\S 5] briefly explains the motivation of this study, giving the physical meaning of definitions and theorems in \S 3 and \S 4, espeically Theorem 4.2.6 claiming the existence of a fully faithful symmetric monoidal functor \[ \mathfrak{Z}:\mathcal{MF}\mathrm{us}_{/\mathcal{E}}\rightarrow\mathcal{BF} \mathrm{us}_{\mathcal{E}}^{\mathrm{cl}} \] where \(\mathcal{MF}\mathrm{us}_{/\mathcal{E}}\) is the category of multi-fusion categories fully faithful over \(\mathcal{E}\) with the equivalence classes of nonzero semisimple bimodules over \(\mathcal{E}\) as morphisms, and \(\mathcal{BF}\mathrm{us}_{\mathcal{E}}^{\mathrm{cl}}\) is the category of nondegenerate braided fusion categories fully faithful containing \(\mathcal{E}\) with the equivalencce classes of closed multi-fusion bimodules containing \(\mathcal{E}\) as morphisms. \end{itemize}
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