We give a rigorous definition of tropical fans (the "local building blocksfor tropical varieties") and their morphisms. For such a morphism of tropicalfans of the same dimension we show that the number of inverse images (countedwith suitable tropical multiplicities) of a point in the target does not dependon the chosen point - a statement that can be viewed as the beginning of atropical intersection theory. As an application we consider the moduli spacesof rational tropical curves (both abstract and in some R^r) together with theevaluation and forgetful morphisms. Using our results this gives new, easy, andunified proofs of various tropical independence statements, e.g. of the factthat the numbers of rational tropical curves (in any R^r) through given pointsare independent of the points.
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