Remarks on Shelah's classification theory and Quillen's negation

TL;DR

This paper reformulates key model-theoretic properties like stability and NIP using category theory, specifically Quillen lifting properties, proposing a homotopy-theoretic approach to model theory.

Contribution

It introduces a novel categorical framework for stability, NIP, NTP, and non-dividing, connecting model theory with homotopy theory through Quillen lifting properties.

Findings

Category-theoretic reformulations of model-theoretic properties

Identification of Quillen lifting properties as characterizations

Proposal of a homotopy-theoretic approach to model theory

Abstract

We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain morphisms associated with linear orders, in a certain category extending the categories of topological spaces and of simplicial sets. This suggests an approach to a homotopy theory for model theory.