# Cellular categories and stable independence

## TL;DR

This paper establishes a connection between cellular categories in algebraic topology and stable independence in model theory, showing their equivalence in certain cases and applying this to stability and tameness of specific classes.

## Contribution

It demonstrates that cellular categories with generated cellular morphisms correspond exactly to those with stable independence, and applies this to prove stability and tameness of certain algebraic classes.

## Key findings

- Cellular categories with generated morphisms are equivalent to those with stable independence.
- Abstract elementary classes of roots of Ext are shown to be stable and tame.
- A simplified proof that combinatorial categories are closed under 2-limits.

## Abstract

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosick\'y.