Throughout this section, we will use the following terminology:
We refer to objects and morphisms of \(\mathrm{Cat}[\mathrm{Set}^{B\mathbb{Z}}]\) as ordinary \(1\)-categories with local shifts and shift-preserving functors.
We refer to objects and morphisms of \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) as \(k\)-linear \((\infty,2)\)-categories with local shifts and shift-preserving \(k\)-linear functors.
We refer to objects and morphisms of \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) as \(k\)-linear stable \((\infty,2)\)-categories with local shifts and shift-preserving \(k\)-linear exact functors.
Similarly, we refer to objects and morphisms in \(\mathrm{Alg}_{\mathbb E_1}\) of the \(\infty\)-categories in (1)–(3) as monoidal ordinary categories with local shifts, etc.