Motivated by an attempt to better understand the notion of a symplectic
stack, we introduce the notion of a symplectic hopfoid, which should be thought
of as the analog of a groupoid in the so-called symplectic category. After
reviewing some foundational material on canonical relations and this category,
we show that symplectic hopfoids provide a characterization of symplectic
double groupoids in these terms. Then, we show how such structures may be used
to produce examples of symplectic orbifolds, and conjecture that all symplectic
orbifolds arise via a similar construction. The symplectic structures on the
orbifolds produced arise naturally from the use of canonical relations. The
characterization of symplectic double groupoids mentioned above is made
possible by an observation which provides various ways of realizing the core of
a symplectic double groupoid as a symplectic quotient of the total space, and
includes as a special case a result of Zakrzewski concerning Hopf algebra
objects in the symplectic category. This point of view also leads to a new
proof that the core of a symplectic double groupoid itself inherits the
structure of a symplectic groupoid. Similar constructions work more generally
for any double Lie groupoid---producing what we call a Lie hopfoid---and we
describe the sense in which a version of the "cotangent functor" relates such
hopfoid structures.Comment: UC Berkeley Ph.D. thesis; 101 page