Tropical fans and the moduli spaces of tropical curves

We give a rigorous definition of tropical fans (the 'local building blocks for tropical varieties') and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R-r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R-r) through given points are independent of the points.