Exercices de style: a homotopy theory for set theory, I

TL;DR

This paper develops a model category framework for set theory, capturing key concepts like finiteness and cardinality, and demonstrates how set-theoretic notions can be interpreted through homotopy theory.

Contribution

It introduces the first model category for set theory based on natural conventions, linking set-theoretic concepts with homotopy theoretic structures.

Findings

Constructed a model category for set theory satisfying key notions.

Revealed set-theoretic concepts like Shelah's covering number via homotopy theory.

Showed the construction follows from standard methods, simplifying verification.

Abstract

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. In a subsequent paper \cite{GaHa1} we give a homotopy theoretic dictionary of set theoretic concepts, most notably Shelah's covering number $\cov(\lambda, \aleph_1,\aleph_1,2)$, recovered from this model category. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verification that the construction works.