Classifying anima of condensed $\infty$-categories of points

TL;DR

This paper compares two classifying anima constructions in condensed $ abla$-categories related to algebraic geometry and model theory, revealing a deep connection between the proétale fundamental group and the Lascar group.

Contribution

It establishes a comparison between classifying anima of two natural condensed $ abla$-categories, linking algebraic geometry and model theory through a unified construction.

Findings

Proves a connection between the proétale fundamental group and the Lascar group.

Shows that both groups are special cases of the same construction.

Provides a new perspective on the relationship between algebraic geometry and model theory.

Abstract

We compare the classifying anima of two natural condensed $\infty$-categories associated to a coherent $\infty$-topos. One from our work with Barwick and Glasman on exit-path categories in algebraic geometry, and the other from Lurie's work on ultracategories. The key consequence of our comparison is a connection between algebraic geometry and model theory: up to a mild completion, the pro\'{e}tale fundamental group of a scheme and the Lascar group of a complete first-order theory are both special cases of the same construction.