  We extend the theory of d-categories, by providing an explicit description of
the right mapping spaces of the d-homotopy category of an $\infty$-category.
Using this description, we deduce an invariant $\infty$-categorical
characterization of the d-homotopy category. We then proceed to develop an
analogous theory of d-operads, which model $\infty$-operads with (d
-1)-truncated multi-mapping spaces, and prove analogous results for them.

Algebraic Topology;Category Theory