  The Reeb graph is a construction which originated in Morse theory to study a
real valued function defined on a topological space. More recently, it has been
used in various applications to study noisy data which creates a desire to
define a measure of similarity between these structures. Here, we exploit the
fact that the category of Reeb graphs is equivalent to the category of a
particular class of cosheaf. Using this equivalency, we can define an
`interleaving' distance between Reeb graphs which is stable under the
perturbation of a function. Along the way, we obtain a natural construction for
smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb
graph can be constructed in polynomial time.
