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ON MINIMAL COALGEBRAS
"... Abstract. We define an outdegree for Fcoalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all Fcoalgebras, this class has a terminal object, which for many problems can stand in for the terminal Fcoalgebra, which need not exist in general. As exam ..."
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Abstract. We define an outdegree for Fcoalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all Fcoalgebras, this class has a terminal object, which for many problems can stand in for the terminal Fcoalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Mooreautomata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and selfcontained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition. For every automaton A there exists a minimal automaton ∇(A), which displays
Physical Systems
, 2009
"... We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimu ..."
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We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics. 1