  We argue for the addition of category theory to the toolkit of toric
topology, by surveying recent examples and applications. Our case is made in
terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric
manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the
homotopy colimit of a diagram of spaces over the small category cat(K), whose
objects are the faces of a finite simplicial complex K and morphisms their
inclusions. Then we study the corresponding cat(K)-diagrams in various
algebraic Quillen model categories, and interpret their homotopy colimits as
algebraic models for X_K. Such models encode many standard algebraic
invariants, and their existence is assured by the Quillen structure. We provide
several illustrative calculations, often over the rationals, including proofs
that quasitoric manifolds (and various generalisations) are rationally formal;
that the rational Pontrjagin ring of the loop space \Omega DJ(K) is isomorphic
to the quadratic dual of the Stanley-Reisner algebra Q[K] for flag complexes K;
and that DJ(K) is coformal precisely when K is flag. We conclude by describing
algebraic models for the loop space \Omega DJ(K) for any complex K, which mimic
our previous description as a homotopy colimit of topological monoids.
