  The notion of an upward plane graph in graph theory and that of a progressive
plane graph (or plane string diagram) in category theory are essentially the
same thing. In this paper, we combine the ideas in graph theory and category
theory to explain why and in what sense upward planarity is a topological
property. The main result is that two upward planar drawings of an acyclic
directed graph are equivalent (connected by a deformation) if and only if they
are connected by a planar isotopy which preserves the orientation and
polarization of $G$. This result gives a positive answer to Selinger's
conjectue, whose strategy is different from the solution recently given by
Delpeuch and Vicary.
