  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.
