Smooth categories in a 6 functor formalism and compact generation for nuclear categories in analytic geometry

TL;DR

This paper explores smooth categories within a six-functor formalism, linking nuclear sheaves on rigid analytic varieties to smoothness and compact generation, using condensed mathematics and analytic stacks.

Contribution

It introduces a new framework connecting smooth $ abla$-categories with nuclear sheaves and demonstrates their properties in analytic geometry.

Findings

Rigid analytic variety is smooth iff its nuclear sheaves category is smooth.

Compact generation of nuclear sheaves relates to algebraization of the variety.

Constructs an example of an internally smooth but non-atomically generated category.

Abstract

In this paper, we study the notion of smooth $\infty$-categories within the framework of a six-functor formalism. By leveraging the theory of condensed mathematics and analytic stacks, we apply these results to demonstrate that a rigid analytic variety is smooth if and only if its associated category of nuclear sheaves is smooth. Furthermore, we relate the compact generation of the category of nuclear sheaves to the algebraization of the rigid analytic variety; these results are then employed to obtain an example of a non atomically generated but internally smooth category.