  As a visualization of Cartier and Foata's "partially commutative monoid"
theory, G.X. Viennot introduced "heaps of pieces" in 1986. These are
essentially labeled posets satisfying a few additional properties. They
naturally arise as models of reduced words in Coxeter groups. In this paper, we
introduce a cyclic version, motivated by the idea of taking a heap and wrapping
it into a cylinder. We call this object a "toric heap", as we formalize it as a
labeled toric poset, which is a cyclic version of an ordinary poset. To define
the concept of a toric extension, we develop a morphism in the category of
toric heaps. We study toric heaps in Coxeter theory, in view of the fact that a
cyclic shift of a reduced word is simply a conjugate by an initial or terminal
generator. This allows us to formalize and study a framework of "cyclic
reducibility" in Coxeter theory, and apply it to model conjugacy. We introduce
the notion of "torically reduced", which is stronger than being cyclically
reduced for group elements. This gives rise to a new class of elements called
"torically fully commutative" (TFC), which are those that have a unique cyclic
commutativity class, and comprise a strictly bigger class than the "cyclically
fully commutative" (CFC) elements. We prove several cyclic analogues of results
on fully commutative (FC) elements due to Stembridge. We conclude with how this
framework fits into recent work in Coxeter groups, and we correct a minor flaw
in a few recently published theorems.
