Infinitary combinatorics in condensed math and strong homology

TL;DR

This paper explores how recent developments in higher derived limits influence various aspects of condensed and pyknotic mathematics, revealing deep combinatorial structures with broad implications across multiple mathematical domains.

Contribution

It provides a unified account of the combinatorics underlying higher derived limits and their impact on sheaf theory, duality, and homological structures in condensed mathematics.

Findings

Implications for sheaf theory of extremally disconnected spaces

Connections to Banach--Smith duality

Insights into the structure of derived categories of condensed abelian groups

Abstract

Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse, pertaining, for example, to the sheaf theory of extremally disconnected spaces, to Banach--Smith duality, to the productivity of compact projective condensed anima, and to the structure of the derived category of condensed abelian groups. Underlying each of these implications are the combinatorics of multidimensionally coherent families of functions of small infinite cardinal height, and it is for this reason that we convene accounts of them together herein.