  Ordered phases resulting from spontaneously broken continuous symmetries are
effectively described by sigma models of maps to the coset space of Goldstone
modes. A classic problem is to classify the topological sectors of the sigma
model. In simple cases, this can be expressed in terms of homotopy groups of
the target space; but in general, it is a complicated affair -- even in two
dimensions -- and a general method is lacking. In this letter, we introduce a
technique to systematically classify topological sectors of sigma models in
various dimensions using a framework based on higher categorical
generalizations of the fundamental group. As a demonstration, we recover some
known results and obtain new ones. The technique and relevant mathematical
structures are described only qualitatively in this letter; details can be
found in our companion paper.
