Recently, the notion of symmetry is generalized so that we can get further information on low-energy dynamics in strongly coupled field theories. This is described in terms of topological operators and also category. It would be crucial that topology of gauge fields is nontrivial in a fully regularized framework; e.g., continuity appears to be lost under lattice regularization. L¥”uscher addressed this issue for $SU(2)$ gauge fields and defined the topological charge on a lattice explicitly. We apply L¥”uscher’s construction to generalized symmetries. We consider lattice $SU(N)$ gauge theories coupled with $¥mathbb{Z}_N$ $ 2$-form gauge fields, and show the fractionality of the topological charge on $SU(N)/¥mathbb{Z}_N$ principal bundle. Also the mixed ‘t Hooft anomaly and higher-group structure are realized. We become interested in monopole physics as a topological phenomenon on the lattice.