nLab
(2,1)-category

By the general rules of $(n,r)$-categories, a $(\mathrm{2,1})$-category is an $\mathrm{\infty }$-category such that

• any $j$-morphism is an equivalence, for $j>1$;
• any two parallel $j$-morphisms are equivalent, for $j>2$.

You can start from any notion of $\mathrm{\infty }$-category, strict or weak; up to equivalence, the result can always be understood as a locally groupoidal $2$-category.