On $d$-categories and $d$-operads

We extend the theory of $d$-categories, which are a strict model for $(d, 1)$-categories introduced by Lurie, 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 doperads, which model $\infty$-operads with $(d-1)$-truncated multimapping spaces, and prove analogous results for them.

Keywords

homotopy theory, infinity-category

2010 Mathematics Subject Classification

18A99, 55Pxx

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The first author was supported by the Alon Fellowship and the ISF grant 1588/18. The second author was supported by the ISF grant 1650/15.

Received 20 February 2019

Received revised 28 May 2019

Accepted 8 June 2019

Published 11 December 2019