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12
Modeling Concurrency with Geometry
, 1991
"... The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer ..."
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Cited by 125 (13 self)
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The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true nfold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude...
Process and Term Tile Logic
, 1998
"... In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis o ..."
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Cited by 33 (25 self)
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In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis of several language implementation efforts, it is useful to map tile logic back into rewriting logic in a conservative way, to obtain executable specifications of tile systems. We extend the results of earlier work by two of the authors, focusing on some interesting cases where the mathematical structures representing configurations (i.e., states) and effects (i.e., observable actions) are very similar, in the sense that they have in common some auxiliary structure (e.g., for tupling, projecting, etc.). In particular, we give in full detail the descriptions of two such cases where (net) processlike and usual term structures are employed. Corresponding to these two cases, we introduce two ca...
Symmetric Monoidal and Cartesian Double Categories as a Semantic Framework for Tile Logic
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2002
"... Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with sideeffects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing acti ..."
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Cited by 13 (9 self)
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Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with sideeffects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing actions (called configurations and observations, respectively, in our terminology) are monoidal categories having the same objects (the interfaces of the system). In particular, configurations and observations based on netprocesslike and term structures are usually described in terms of symmetric monoidal and cartesian categories, where the auxiliary structures for the rearrangement of interfaces correspond to suitable natural transformations. In this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (for example, for configurations or for observations, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four double functors (rather than two). As a consequence, the concepts of symmetric monoidal and cartesian (with consistently chosen products) double categories arise in a natural way from the corresponding ordinary versions, giving a very good relationship between the auxiliary structures of configurations and observations. Moreover, the Kelly–Mac Lane coherence axioms can be lifted to our setting without effort, thanks to the characterization of two suitable diagonal categories that are always present in a double category. Then, symmetric monoidal and cartesian double categories are shown to offer an adequate semantic setting for process and term tile systems.
Symmetric and Cartesian Double Categories as a Semantic Framework for Tile Logic
, 1995
"... this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (e.g., for configurations or for eff ..."
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Cited by 6 (5 self)
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this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (e.g., for configurations or for effects, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four
Lie 2algebras
, 2004
"... I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times ..."
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Cited by 4 (1 self)
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I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times during my graduate studies. I also thank my ‘mathematical brothers’: Miguel CarriónÁlvarez, Toby Bartels, Jeffrey Morton, and Derek Wise for their friendship and engaging, educational conversations. I am grateful for the assistance of Aaron Lauda in drawing various braid diagrams, and thank Ronnie Brown, Andrée Ehresmann, Thomas Larsson, James Stasheff, J. Scott Carter, and Masahico Saito for helpful discussions and correspondence. Finally, I am extremely appreciative of the love and support of my family, friends, and former professors during my time as a graduate student. I certainly could not have accomplished all that I have without them. iii ABSTRACT OF THE DISSERTATION
Extended TQFT’s and Quantum Gravity
, 2007
"... Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topolo ..."
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Cited by 2 (1 self)
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Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the DijkgraafWitten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a highercategorical version of Vect, denoted 2Vect, a bicategory of 2vector spaces. Along the way, we prove several results showing how to construct 2vector spaces of Vectvalued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2morphisms in 2Vect for the extended TQFT, and that these
Abstract Modeling Concurrency with Geometry
"... The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer ..."
Abstract
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The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true nfold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude with a formal definition of higher dimensional automaton as an ncomplex or ncategory, whose two essential axioms are associativity of concatenation within dimension and an interchange principle between dimensions. 1
A 2category View for Double Categories with Shared Structure
, 1999
"... 2categories and double categories are respectively the natural semantic ground for rewriting logic (rl) and tile logic (tl). Since 2categories can be regarded as a special case of double categories, then rl can be easily embedded into tl, where also rewriting synchronization is considered. Since ..."
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2categories and double categories are respectively the natural semantic ground for rewriting logic (rl) and tile logic (tl). Since 2categories can be regarded as a special case of double categories, then rl can be easily embedded into tl, where also rewriting synchronization is considered. Since rl is the semantic basis of several existing languages, it is useful to map tl back into rl to have an executable framework for tile specifications. We extend the results of a previous work of two of the authors, focusing on tile systems where the algebraic structures for configurations and observations rely on some common auxiliary structure (e.g., for pairing, projecting, etc.). The new model theory required to relate the categorical models of the two logics is an extended version of the theory of 2categories, and is defined using partial membership equational logic. More concretely, this semantic mapping yields a rewriting logic implementation of tile logic, where a metalayer is requir...
Double categories, 2categories . . .
 THEORY APPL. CATEG
, 1999
"... The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2categories is equivalent to the category of edg ..."
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The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2categories is equivalent to the category of edge symmetric double categories with thin structure.