In a one-dimensional space, any nullhomotopic loop factors through a dendrite. Analogously, we can say that two paths $f$ and $g$, with the same endpoints, are equivalent if $f*\overline g$ factors through a loop in a dendrite, where $\overline g$ is the path $g$ traversed backwards. We will show that the equivalence relation generated by this relation is the same as the path-homotopy and discuss its consequences.

This is joint work with Greg Conner, Jeremy Brazas, and Paul Fabel.