Locally rigid $\infty$-categories

TL;DR

This work develops a comprehensive framework for locally rigid and rigid symmetric monoidal ∞-categories relative to a base $\mathcal{V}$. It introduces $\mathcal{V}$-atomic and trace-class morphisms to compare rigidity notions and proves that any locally rigid commutative $\mathcal{V}$-algebra $\mathcal{W}$ admits a rigidification $\mathrm{Rig}_{\mathcal{V}}(\mathcal{W})$, with a fully faithful left adjoint embedding into $\mathcal{W}$. The paper establishes presentability for rigid algebras and provides explicit (Sp-centric) and general constructions of rigidifications, including an exceptional functor $\Gamma_!$ that encodes the duality data. The results connect to Clausen–Scholze’s nuclear modules in the $\mathcal{V}=\mathbf{Sp}$ setting and extend rigidity theory to completions (localizations and local limits) of rigid categories, offering a robust toolkit for tt-geometry beyond compact generation. Overall, the work unifies rigidity, atomicity, and weighted colimits to describe completions of rigid ∞-categories and their module theories in broad geometric and algebraic contexts.

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 $\mathcal{V}$.

Theorems & Definitions (257)

• Theorem A
• Theorem B
• Theorem C
• Theorem 2.1: heine
• Definition 1
• Definition 2
• Remark 1
• Example 1
• Example 2
• Corollary 1