Super duality and Crystal bases for quantum orthosymplectic superalgebras

We introduce a semisimple tensor category $\text{mc}{O}_q^{int}(m\mathrm{\mid }n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{m+n}$, $C_{m+n}$, $D_{m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\text{mc}{O}_q^{int}(m\mathrm{\mid }n)$ and show that an irreducible module in $\text{mc}{O}_q^{int}(m\mathrm{\mid }n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatorial object called ortho-symplectic tableaux.