Locally rigid $\mathrm{\infty }$-categories

We develop the theory of locally rigid and rigid symmetric monoidal $\mathrm{\infty }$-categories over an arbitrary base $\mathcal{V}\in \mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}({\mathbf{P}\mathbf{r}}^{\mathrm{L}})$. Among other things, we prove that every locally rigid commutative $\mathcal{V}$-algebra arises as a completion'' of a rigid commutative $\mathcal{V}$-algebra. Along the way, we introduce and study $\mathcal{V}$-atomic morphisms’', which are analogues of compact morphisms over an arbitrary base $\mathcal{V}$.