Title: The classical world from quantum theory
Abstract: We consider symmetric monoidal dagger-categories with classical objects, that is, with dagger-Frobenius algebras. Such categories have been identified to capture important fragments of quantum behavior and, in particular, also quantum-classical interaction including measurements. From any such category we recover important subcategories namely: the (classical) probabilistic cone, partial functions, relations, (doubly) stochastic maps, total functions etc. We obtain an analysis of how two distinct instances of classical probability live within the quantum structure: superposition & mixture. We also recover important results from quantum informatics, e.g. Nielsen's preorder on bipartite entangled states, Naimarks theorem, dense coding etc.