  An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The
restriction F|_A: A to X of an action is called a cyclic map. Special cases of
these notions include group actions and the Gottlieb groups of a space, each of
which has been studied extensively. We prove some general results about actions
and their Eckmann-Hilton duals. For instance, we classify the actions on an
H-space that are compatible with the H-structure. As a corollary, we prove that
if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) =
Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on
Omega(X). We introduce a new notion of the category of a map g and prove that g
is cocyclic if and only if the category is less than or equal to 1. From this
we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <=
1. We also briefly discuss the relationship between a map being cyclic and its
cocategory being <= 1.
