Exercices de style: A homotopy theory for set theory II

TL;DR

This paper develops a homotopy-theoretic framework for set theory, revealing that homotopy invariants correspond to key combinatorial set-theoretic concepts like covering numbers and revised power functions.

Contribution

It introduces a model category for set theory and demonstrates how homotopy invariants relate to important set-theoretic functions, providing new conceptual tools.

Findings

Homotopy-invariant cardinality corresponds to Shelah's covering number.

Revised power functions can be derived using homotopy-theoretic methods.

A set-theoretic dictionary in the model category aids in understanding set theory through homotopy concepts.

Abstract

This is the second part of a work initiated in \cite{GaHa}, where we constructed a model category, $\Qt$, for set theory. In the present paper we use this model category to introduce homotopy-theoretic intuitions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah's PCF theory, and that other combinatorial objects, such as Shelah's revised power function - the cardinal function featuring in Shelah's revised GCH theorem - can be obtained using similar tools. We include a small "dictionary" for set theory in $\QtNaamen$, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.