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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
Coalgebras, Chu Spaces, and Representations of 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.
Presentation of set functors: a coalgebraic perspective
"... Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1. ..."
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Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1.
UNIVERSAL COALGEBRAS AND THEIR LOGICS
, 2009
"... ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1. ..."
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ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.
ON COALGEBRAS AND TYPE TRANSFORMATIONS
"... Abstract. We show that for an arbitrary Setendofunctor T the generalized membership function given by a subcartesian transformation µ from T to the filter functor F can be alternatively defined by the collection of subcoalgebras of constant Tcoalgebras. Subnatural transformations ε between any t ..."
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Abstract. We show that for an arbitrary Setendofunctor T the generalized membership function given by a subcartesian transformation µ from T to the filter functor F can be alternatively defined by the collection of subcoalgebras of constant Tcoalgebras. Subnatural transformations ε between any two functors S and T are shown to be subcartesian if and only if they respect µ. The class of Tcoalgebras whose structure map factors through ε is shown to be a covariety if ε is a natural and subcartesian monotransformation. 1. SetFunctors Our interest in SetFunctors arises from their use as signatures of algebras or coalgebras. A Setfunctor T associates with each set X a set T (X) and with each map f: X → Y between sets a map T f: T (X) → T (Y) so that identities and function compositions are preserved, i.e. T idX = id T (X) and T (g ◦ f) = T g ◦ T f whenever f: X → Y and g: Y → Z. In the context of universal algebra, the most important examples are given by the so called polynomial functors. Starting with a sequence of natural numbers
Abstract CMCS 2006 Modularity in Coalgebra
"... This paper gives an overview of recent results concerning the modular derivation of (i) modal specification logics, (ii) notions of simulation together with logical characterisations, and (iii) sound and complete axiomatisations, for systems modelled as coalgebras of functors on Set. Our approach ap ..."
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This paper gives an overview of recent results concerning the modular derivation of (i) modal specification logics, (ii) notions of simulation together with logical characterisations, and (iii) sound and complete axiomatisations, for systems modelled as coalgebras of functors on Set. Our approach applies directly to an inductivelydefined class of coalgebraic types, which subsumes several types of discrete statebased systems, including (probabilistic) transition systems, probabilistic automata and spatial transition systems.