Categorical notions of layered tropical algebra and geometry

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories, together with a "tropicalization functor" which permits us to pass from usual algebraic geometry to the tropical world. We consider tropical varieties from this categorical viewpoint, with emphasis on polynomial functions and their roots.