# On d-Categories and d-Operads

## TL;DR

This paper extends the theory of d-categories and d-operads by providing explicit descriptions of their mapping spaces and invariants, advancing the understanding of higher categorical structures.

## Contribution

It offers an explicit description of right mapping spaces in d-homotopy categories and develops a theory of d-operads with truncated multi-mapping spaces.

## Key findings

- Explicit description of right mapping spaces in d-homotopy categories
- Invariant characterization of d-homotopy categories
- Development of d-operads with truncated multi-mapping spaces

## Abstract

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.