Locally rigid $\infty$-categories

TL;DR

This paper develops the theory of locally rigid and rigid symmetric monoidal ∞-categories over arbitrary bases, proving that locally rigid commutative algebras can be obtained as completions of rigid ones, and introduces new concepts like V-atomic morphisms.

Contribution

It introduces the theory of locally rigid ∞-categories over arbitrary bases and shows how locally rigid commutative algebras relate to rigid ones, along with new morphism concepts.

Findings

Every locally rigid commutative V-algebra is a completion of a rigid algebra.

Introduction and analysis of V-atomic morphisms as analogues of compact morphisms.

Development of the theory over arbitrary base 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 $\mathcal{V}$.