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Notes on enriched categories with colimits of some class
 Theory Appl. Categ
"... The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are ..."
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The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are those ψ for which ψlimits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of [AK88]). We prove that for the class P of all weights, the classes P + and P − both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φcocompletion Φ(A) of A; and we recognize Q(A) as the Cauchycompletion of A. We study the equivalence between (Q(A op)) op and Q(A), which we exhibit as the restriction of the Isbell adjunction between [A, V] op and [A op, V] when A is small; and we give a new Morita theorem for any class Φ containing Q. We end with the study of Φcontinuous weights and their relation to the Φflat weights. 1
Coalgebraic semantics for timed processes
 Inf. & Comp
, 2006
"... We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comon ..."
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Cited by 8 (1 self)
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We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws 1
More On Injectivity In Locally Presentable Categories
, 2002
"... Injectivity with respect to morphisms having #presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, #directed colimits, and #pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 78 ..."
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Cited by 7 (3 self)
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Injectivity with respect to morphisms having #presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, #directed colimits, and #pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785804). In contrast, for geometric logic an example is found of a class closed under directed colimits and pure subobjects, but not axiomatizable by a geometric theory. A more technical characterization of axiomatizable classes in geometric logic is presented. 1.
A note on the free regular and exact completions and their infinitary generalizations, Theory and Applications of Categories
, 1996
"... ABSTRACT. Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory. ..."
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ABSTRACT. Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.
HOMOTOPICAL INTERPRETATION OF GLOBULAR COMPLEX BY MULTIPOINTED DSPACE
"... Abstract. Globular complexes were introduced by E. Goubault and the author to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CWcomplex. We prove that there exists a ..."
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Abstract. Globular complexes were introduced by E. Goubault and the author to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CWcomplex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows. The underlying
More on Orthogonality in Locally Presentable Categories
"... Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category L, representing the signature in some sense, is given, together with a set \Sigma of morphisms having finitely presentable domains and codomains. And our category K ..."
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Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category L, representing the signature in some sense, is given, together with a set \Sigma of morphisms having finitely presentable domains and codomains. And our category K is the full subcategory of L on all objects K orthogonal to ) Supported by the Grant Agency of the Czech Republic under the grant No. 201/99/0310. 1 each s : X ! X 0 in \Sigma (notation: K ? s), which means that every morphism f : X ! K uniquely factors through s; notation: K = \Sigma ? . Such subcategories K of L are called in [AR] the !orthogonality c
COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION
"... Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large n ..."
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Cited by 2 (2 self)
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Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choiceofduals functor on the compact
Automata theory in nominal sets
, 2012
"... Abstract. We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we ..."
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Abstract. We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.
On Data Types Presented By Multiequations
, 2001
"... Equational presentation of abstract data types is generalized to presentation by multiequations, i.e., exclusiveor's of equations, in order to capture parametric data types such as array or set. Multiinitialalgebra semantics for such data types is introduced. Classes of algebras described by multi ..."
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Equational presentation of abstract data types is generalized to presentation by multiequations, i.e., exclusiveor's of equations, in order to capture parametric data types such as array or set. Multiinitialalgebra semantics for such data types is introduced. Classes of algebras described by multiequations are characterized.
Continuous Categories Revisited
, 2003
"... Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of ..."
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Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal. 1.