I shall talk about Reeb graphs, and the benefits of regarding them as set-valued cosheaves on the real line. For instance, there is a smoothing operation and an interleaving distance for these cosheaves, which can be interpreted as a smoothing operation and an interleaving distance for Reeb graphs. This is joint work with Elizabeth Munch and Amit Patel. If time allows, I will discuss various extensions of these ideas, worked out with the additional collaboration of Anastasios Stefanou; and also of Pomona College undergraduates Song Yu and Dmitriy Smirnov.