On \(d\)-categories and \(d\)-operads (Q2284289)

A strict \(d\)-category, in the sense of Lurie, is an \(\infty\)-category \(\mathcal C\) such that maps \(\partial \Delta^d \to \mathcal C\) admit at most one possible extension to \(\Delta^d\) up to homotopy and for \(m>d\), a map \(\Delta^m\to \mathcal C\) is completely determined by its value on the boundary \(\partial \Delta^m\). One can homotopy relax this definition to the one of a \textit{essentially \(d\)-category}, by requiring instead that for all \(X,Y \in \mathcal C\), the mapping space \(\mathrm{Map}_{\mathcal C}(X,Y)\) to be \(d-1\)-truncated, i.e. \(\pi_{\geq d}(\mathrm{Map}_{\mathcal C}(X,Y)) = 0\). For instance, an essential \(1\)-category is an \(\infty\)-category in the essential immage of the nerve functor \(N:\mathrm{Cat}\to \mathrm{Cat}_\infty\), while a \(1\)-category must be \textit{isomorphic} to the nerve of an ordinary category. There exists a canonical way to truncate an \(\infty\)-category \(\mathcal C\) to a \(d\)-category \(h_d\mathcal C\). The goal of this paper is to describe the relations between \(\mathcal C\) and \(h_d\mathcal C\). For this, the authors show that \(h_d\) can be promoted to a left adjoint of the inclusion \(\mathrm{Cat}_d\hookrightarrow \mathrm{Cat}_\infty\) of essential \(d\)-categories inside of \(\infty\)-categories and that the \(d\)-category \(h_d\mathcal C\) is obtained by \((d-1)\)-truncation of the mapping spaces of \(\mathcal C\). Furthermore, the unit of the adjunction \(\mathcal C \to h_d\mathcal C\) is essentially surjective. In the last section, the authors show that a similar theory holds for \(\infty\)-operads. By imposing similar restrictions, one has a notion of essential \(d\)-operad. By defining the \(d\)-homotopy operad functor, the authors show that the inclusion of essential \(d\)-operads in \(\infty\)-operads admit a left adjoint like \(h_d\) whose unit is essentially surjective and whose multimapping are given by \((d-1)\)-truncation.