  Sigma models effectively describe ordered phases of systems with
spontaneously broken symmetries. At low energies, field configurations fall
into solitonic sectors, which are homotopically distinct classes of maps.
Depending on context, these solitons are known as textures or defect sectors.
In this paper, we address the problem of enumerating and describing the
solitonic sectors of sigma models. We approach this problem via an algebraic
topological method -- combinatorial homotopy, in which one models both
spacetime and the target space with algebraic objects which are higher
categorical generalizations of fundamental groups, and then counts the
homomorphisms between them. We give a self-contained discussion with plenty of
examples and a discussion on how our work fits in with the existing literature
on higher groups in physics.
