Generalized Yetter-Drinfel'd module categories for regular multiplier Hopf algebras

For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel’d module category ${}_A{\mathcal{Y}\mathcal{D}}^A$ is equivalent to the centre $Z({}_A\mathcal{M})$ of the unital left $A$-module category ${}_A\mathcal{M}$. Then we introduce the generalized $(\alpha, \beta)$-Yetter-Drinfe{l}^{\mathrm{\prime }}dmodulecategories${}_{A}$\mathcal{G}\mathcal{Y}\mathcal{D}$^{A}(\alpha, \beta)$,whicharetreatedascomponentsofabraided$T$-category. Especially when $A$ is a coFrobenius Hopf algebra, ${}_A{\mathcal{Y}\mathcal{D}}^A(\alpha ,\beta )$ is isomorphic to the unital $\widehat{A}\bowtie A(\alpha ,\beta )$-module category ${}_{\widehat{A}\bowtie A(\alpha ,\beta )}\mathcal{M}$. Finally for a Yetter-Drinfel’d $A$-module algebra $H$, we introduce Yetter-Drinfel’d $(H, A)$-module category, which is a monoidal.