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24
Higher gauge theory
"... Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where t ..."
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Cited by 41 (12 self)
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Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2groups and Lie 2algebras. We describe a theory of 2connections on principal 2bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2connection allows parallel transport along paths and surfaces in a parametrizationindependent way. In terms of Breen and Messing’s framework, this requires that the ‘fake curvature ’ must vanish. In this paper we summarize the main results of our theory without proofs. 1
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 18 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Graph Rewriting, Constraint Solving and Tiles for Coordinating Distributed Systems
 Applied Categorical Structures
, 1999
"... . In this paper we describe an approach to model the dynamics of distributed systems. For distributed systems we mean systems consisting of concurrent processes communicating via shared ports and posing certain synchronization requirements, via the ports, to the adjacent processes. The basic idea is ..."
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Cited by 17 (14 self)
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. In this paper we describe an approach to model the dynamics of distributed systems. For distributed systems we mean systems consisting of concurrent processes communicating via shared ports and posing certain synchronization requirements, via the ports, to the adjacent processes. The basic idea is to use graphs to represent states of such systems, and graph rewriting to represent their evolution. The kind of graph rewriting we use is based on simple contextfree productions which are however combined by means of a synchronization mechanism. This allows for a good level of expressivity in the system without sacrifying full distribution. To formally model this kind of graph rewriting, however, we do not adopt the classical graph rewriting style but a more general framework, called the tile model, which allows for a clear separation between sequential rewriting and synchronization. Then, since the problem of satisfying the synchronization requirements may be a complex combinatorial pro...
Mackaay: Categorical representations of categorical groups
, 2004
"... The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1 ..."
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Cited by 15 (0 self)
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The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1
Tiles, Rewriting Rules and CCS
"... In [12] we introduced the tile model, a framework encompassing a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting and of concurrency theory, and our formalism recollects many properties ..."
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Cited by 14 (8 self)
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In [12] we introduced the tile model, a framework encompassing a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting and of concurrency theory, and our formalism recollects many properties of these sources. For example, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Moreover, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and sideeffects to determine the actual behaviour of a system. In this work we narrow our scope, presenting a restricted version of our tile model and focussing our attention on its expressive power. To this aim, we recall the basic definitions of the process algebras paradigm [3,24], centering the paper on the recasting of this framework in our formalism.
Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
"... This document was typeset using L ..."
MONADS AND COMONADS ON MODULE CATEGORIES
"... known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodu ..."
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Cited by 12 (10 self)
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known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B, −) and HomA(C, −) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find
Enriched Categories as Models of Computation
 in Proc. Fifth Italian Conference on Theoretical Computer Science, ICTCS'95 , World Scientific
, 1996
"... . In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we ela ..."
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Cited by 11 (4 self)
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. In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we elaborate a twosteps procedure allowing for the description of all the sequences of transitions performed by a given system, and equipping them with a suitable equivalence relation. This relation provides the sistem under analisys with a concurrent semantics: equivalence classes denote families of "computationally equivalent" behaviours, corresponding to the execution of the same set of (causally) independent transition steps. 1 Introduction The latest years have seen a wide amount of different approaches to the semantics of computional sistems: a variety that, if only for the comparison between the various formalisms, calls for a unified framework. In this paper we aim to show that enriched ...
CPO models for infinite term rewriting
 in Proc. AMAST'95, LNCS 936
, 1995
"... . Infinite terms in universal algebras are a wellknown topic since the seminal work of the ADJ group [1]. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic gra ..."
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Cited by 10 (7 self)
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. Infinite terms in universal algebras are a wellknown topic since the seminal work of the ADJ group [1]. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic graph is a finitary description of a possibly infinite term. In this paper we introduce infinite rewriting logic, working on the framework of rewriting logic proposed by Jos'e Meseguer [13, 14]. We provide a simple algebraic presentation of infinite computations, recovering the infinite parallel term rewriting, originally presented by one of the authors ([6]) to extend the classical, settheoretical approach to tr with infinite terms. Moreover, we put all the formalism on firm theoretical bases, providing (for the first time, to the best of our knowledge, for infinitary rewriting systems) a clean algebraic semantics by means of (internal) 2categories. 1 Introduction Term rewriting sy...