Recent work on classification problems arising in one-dimensional complex analytic dynamics suggest an underlying theory of mapping class semigroups. Let $S^2$ denote the two-sphere, and fix a finite set $P subset S^2$. The set $BrCov(S^2, P)$ of orientation-preserving branched covering maps of pairs $f: (S^2, P) to (S^2, P)$ of degree at least two and whose branch values lie in $P$ is closed under composition and under pre- and post-composition by orientation-preserving homeomorphisms $h: (S^2, P) to (S^2, P)$ fixing $P$ set wise. Composition descends to a well-defined map on homotopy classes relative to $P$, yielding a countable semigroup $BrMod(S^2, P)$. In addition to the semigroup structure, $BrMod(S^2, P)$ is naturally equipped with two commuting actions of the mapping class group $Mod(S^2, P)$, induced by pre- and post-composition. This richer {em biset} structure, and a related circle of constructions, turn out to be extraordinarily useful in this context. They lead to: algebraic invariants of elements of $BrCov(S^2, P)$ and an analog of the Baer-Dehn-Nielsen theorem; analogs of classical Hurwitz classes; conjugacy invariants; canonical decompositions and forms; an analog of Thurston's trichotomous classification of mapping classes; and induced dynamics on Teichm"uller spaces. This talk is based on algebraic and dynamical perspectives growing out of work of L. Bartholdi and V. Nekrashevych, and is based on ongoing conversations with S. Koch, D. Margalit, and N. Selinger. Not being an expert in topology, I invite your comments and welcome your thoughts.
