  We describe a new sequence of polytopes which characterize A_infinity maps
from a topological monoid to an A_infinity space. Therefore each of these
polytopes is a quotient of the corresponding multiplihedron. Later term(s) in
our sequence of polytopes are demonstrated not to be combinatorially equivalent
to the associahedron, as was previously assumed. They are given the new
collective name composihedra. We point out how these polytopes are used to
parameterize compositions in the formulation of the theories of enriched
bicategories and pseudomonoids in a monoidal bicategory. We present a simple
algorithm for determining the extremal points in Euclidean space whose convex
hull is the nth polytope in the sequence of composihedra, that is, the nth
composihedron.
