ScalingStacks

Definition 3.6. Given an $\infty$-operad $p\colon\mathcal{O}^{\otimes}\to\mathbf{Fin}_{*}$, we define its $d$-homotopy operad $h_{d}\mathcal{O}$ to be a map of simplicial sets $p_{d}\colon\left(h_{d}\mathcal{O}\right)^{\otimes}\to\mathbf{Fin}_{*}$ defined as follows:

1. (1)

   For $d\geq 1$, we simply apply $h_{d}$ to $p$ as a functor between $\infty$-categories and use the fact that $\mathbf{Fin}_{*}$ is a $1$-category; hence there is a canonical isomorphism $h_{d}\left(\mathbf{Fin}_{*}\right)\simeq\mathbf{Fin}_{*}$.

2. (2)

   For $d=0$, we first construct the (ordinary) category $\tilde{h}_{0}\mathcal{O}^{\otimes}$ whose objects are those of $\mathcal{O}^{\otimes}$ and each mapping space is replaced by its image in $\mathbf{Fin}_{*}$. Then we identify isomorphic objects in $\tilde{h}_{0}\mathcal{O}^{\otimes}$ (note that there is a unique induced composition, since isomorphic objects are mapped to the same object in $\mathbf{Fin}_{*}$) and finally we define $\left(h_{0}\mathcal{O}\right)^{\otimes}$ to be the nerve of the resulting category, with $p_{0}$ being the obvious map to $\mathbf{Fin}_{*}$.

3. (3)

   For $d=-1$, we define $p_{d}\colon\mathbf{Fin}_{*}\to\mathbf{Fin}_{*}$ to be the identity functor if $\mathcal{O}^{\otimes}\neq\varnothing$ and the unique functor $p_{d}\colon\varnothing\to\mathbf{Fin}_{*}$ otherwise.