Fractured Structures in Condensed Mathematics

TL;DR

This paper introduces a fractured structure on the $ abla$-topos of condensed anima, providing new insights into its properties and explicitly identifying conservative points, while also analyzing limits in extremally disconnected spaces.

Contribution

It constructs a fractured structure on condensed anima and analyzes limits in extremally disconnected spaces to answer a key question from Clausen.

Findings

Explicit collection of jointly conservative points for condensed anima

Demonstration that the category does not admit all fibers

Construction of a fractured structure in the sense of Lurie

Abstract

We construct a fractured structure, in the sense of Lurie, on the $\infty$-topos of condensed anima. This fractured structure allows us to better comprehend various properties of condensed anima - we use it to exhibit an explicit collection of jointly conservative points for condensed anima. To rule out further candidates for fractured structures, we analyze limits in the category of extremally disconnected spaces. In particular, we show that it does not admit all fibers, answering a question from Clausen.