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36
The Virtues of Etaexpansion
, 1993
"... Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are li ..."
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Cited by 36 (4 self)
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Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fijnormal forms) can be recovered by "cutting" the loops.
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Semantical Principles in the Modal Logic of Coalgebraic
"... Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natur ..."
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Cited by 30 (6 self)
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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.
Expressive Logics for Coalgebras via Terminal Sequence Induction
 Notre Dame J. Formal Logic
, 2002
"... This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras. ..."
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Cited by 26 (8 self)
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This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras.
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, an ..."
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Cited by 17 (1 self)
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A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
Automata and fixed point logics: a coalgebraic perspective
 Electronic Notes in Theoretical Computer Science
, 2004
"... This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton a ..."
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Cited by 15 (7 self)
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This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite twoplayer graph game. We also introduce a language of coalgebraic fixed point logic for Fcoalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an Fautomaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed Fcoalgebras in which p holds.
Coalgebraic Structure From Weak Limit Preserving Functors
, 1999
"... Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and ..."
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Cited by 13 (7 self)
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Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and subcoalgebras. We give a characterization of monos in Set F in terms of congruences and bisimulations, which explains, under which conditions monos must be injective maps.
Contextfree languages via coalgebraic trace semantics
 International Conference on Algebra and Coalgebra in Computer Science (CALCO’05), volume 3629 of Lect. Notes Comp. Sci
, 2005
"... Abstract. We show that, for functors with suitable mild restrictions, the initial algebra in the category of sets and functions gives rise to the final coalgebra in the (Kleisli) category of sets and relations. The finality principle thus obtained leads to the finite trace semantics of nondeterminis ..."
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Cited by 11 (8 self)
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Abstract. We show that, for functors with suitable mild restrictions, the initial algebra in the category of sets and functions gives rise to the final coalgebra in the (Kleisli) category of sets and relations. The finality principle thus obtained leads to the finite trace semantics of nondeterministic systems, which extends the trace semantics for coalgebras previously introduced by the second author. We demonstrate the use of our technical result by giving the first coalgebraic account on contextfree grammars, where we obtain generated contextfree languages via the finite trace semantics. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures. 1
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.