A Note on Braided $T$-categories over Monoidal Hom-Hopf Algebras

Let $Au{t}_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{M}\mathcal{H}\mathcal{D}(H)$ and prove that the $T$-category $Rep(\mathcal{M}\mathcal{H}\mathcal{D}(H))$ of representation of $\mathcal{M}\mathcal{H}\mathcal{D}(H)$ is isomorphic to $\mathcal{M}\mathcal{H}\mathcal{Y}\mathcal{D}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $\mathcal{Z}\mathcal{M}\mathcal{H}\mathcal{Y}\mathcal{D}(H)$ over $\mathbb{Z},$ generalizing the main construction by Staic \cite{S2007}.