Definition [efr-002W]

Let \mathcal {M} act on \mathcal {C}. Then we denote by \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}) the double category given by Proposition [efr-002Y] applied to \mathsf {Para}_\mathcal {M}(\mathcal {C})_* \hookrightarrow \mathsf {Para}_\mathcal {M}(\mathcal {C}), where \mathsf {Para}_\mathcal {M}(\mathcal {C}) is the corresponding locally graded category as in Proposition [efr-002V], and (-)_* denotes the fiber over the basepoint as in Proposition [efr-002X].