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Guarded Transitions in Evolving Specifications
 Proceedings of AMAST 2002, volume 2422 of LNCS
, 2002
"... We represent state machines in the category of specifications, where assignment statements correspond exactly to interpretations be tween theories [7, 8]. However, the guards on an assignment require a special construction. In this paper we raise guards to the same level as assignments by treat ..."
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Cited by 11 (7 self)
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We represent state machines in the category of specifications, where assignment statements correspond exactly to interpretations be tween theories [7, 8]. However, the guards on an assignment require a special construction. In this paper we raise guards to the same level as assignments by treating each as a distinct category over a shared set of objects. A guarded assignment is represented as a pair of arrows, a guard at'row and an assignment arrow. We give a general construction for combining at'rows over a factorization system, and show its specialization to the category of specifications. This construction allows us to define the fine structure of state machine morphisms with respect to guards.
The Euler Characteristic of a Category
 DOCUMENTA MATH.
, 2008
"... The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generali ..."
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Cited by 10 (3 self)
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The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generalization of Rota’s Möbius inversion from posets to categories.
Monads And Interpolads In Bicategories
, 1997
"... . Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The local ..."
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Cited by 8 (4 self)
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. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchycomplete, but have a wellknown Cauchycompletion in common. This prompts us to formulate a concept of Cauchycompleteness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo1cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo1cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...
Generalized operads and their inner cohomomorphisms, arXiv:math.CT/ 0609748
, 2006
"... Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that the ..."
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Cited by 8 (1 self)
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Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =
Coalgebras of Bounded Type
"... Using results of Trnkov'a, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal F coalgebra. Several equivalent characterizations of boundedness are provided. ..."
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Cited by 8 (4 self)
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Using results of Trnkov'a, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal F coalgebra. Several equivalent characterizations of boundedness are provided.
Quotients of the multiplihedron as categorified associahedra
 Homotopy, Homology and Appl
, 2008
"... Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associah ..."
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Cited by 7 (2 self)
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Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
Generalized ultrametric spaces in quantitative domain theory
 Theoretical Computer Science
"... Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be ..."
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Cited by 6 (0 self)
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Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be defined for every metric space and that reflects many of its properties. On the other hand, in order to obtain a broader framework for applications and possible connections to domain theory, generalized ultrametric spaces (gums) have been introduced. In this paper, we employ the space of formal balls as a tool for studying these more general metrics by using concepts and results from domain theory. It turns out that many properties of the metric can be characterized by conditions on its formalball space. Furthermore, we can state new results on the topology of gums as well as two modified fixed point theorems, which may be compared to the PrießCrampe and Ribenboim theorem and the Banach fixed point theorem, respectively. Deeper insights into the nature of formalball spaces are gained by applying methods from category theory. Our results suggest that, while being a useful tool for the study of gums, the space of
A cartesian closed category of approximable concept structures
 Proceedings of the International Conference On Conceptual Structures
, 2004
"... Abstract. Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may offer connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection betwe ..."
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Cited by 6 (4 self)
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Abstract. Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may offer connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in [25]. Building on a new notion of approximable concept introduced by Zhang and Shen [26], this paper provides an appropriate notion of morphisms on formal contexts and shows that the resulting category is equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings. Since the latter categories are cartesian closed, we obtain a cartesian closed category of formal contexts that respects both the context structures as well as the intrinsic notion of approximable concepts at the same time. 1
Associative structures based upon a categorical braiding, preprint math.CT/0512165
, 2005
"... ABSTRACT. It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2category VCat of enriched categories and functors over V, a monoidal bicategory VMod of enriched categories and modules, a catego ..."
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Cited by 5 (4 self)
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ABSTRACT. It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2category VCat of enriched categories and functors over V, a monoidal bicategory VMod of enriched categories and modules, a category of operads in V and a 2fold
A categorical view of algebraic lattices in formal concept analysis
 Fundamenta Informaticae
, 2006
"... Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or represent ..."
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Cited by 4 (4 self)
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Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a categorytheoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating wellknown structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory. 1