Locally rigid $\infty$-categories

Abstract

We develop the theory of locally rigid and rigid symmetric monoidal $\infty$-categories over an arbitrary base $\mathcal{V}\in \mathrm{CAlg}({\mathbf{Pr}}^{\mathrm{L}})$. 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 $\mathcal{V}$.

59 pages, comments welcome! v2: Corrected the (wrong) claim that a certain category of locally rigid categories is presentable after a remark by Jiacheng Liang (the rigid case is unaffected), otherwise minor modifications