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29
Symmetry Breaking in Anonymous Networks: Characterizations
, 1996
"... We characterize exactly the cases in which it is possible to elect a leader in an anonymous network of processors by a deterministic algorithm, and we show that for every network there is a weak election algorithm (i.e., if election is impossible all processors detect this fact in a distributed way) ..."
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Cited by 35 (10 self)
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We characterize exactly the cases in which it is possible to elect a leader in an anonymous network of processors by a deterministic algorithm, and we show that for every network there is a weak election algorithm (i.e., if election is impossible all processors detect this fact in a distributed way). 1 Introduction We consider the problem of electing a leader in an anonymous network of processors. More precisely our model is that of a directed graph, with vertices corresponding to processors, and arcs to communication links (we freely interchange symmetric digraphs and undirected graphs). We make no assumption on the structure of the network: selfloops and parallel arcs are allowed. In particular, processors are anonymous: they do not have unique identifiers. We consider both synchronous and asynchronous processor activation models, and models with and without "port awareness" (local names for outgoing and/or for incoming arcs). We consider both unidirectional and bidirectional links. ...
Fibrations of Graphs
 DISCRETE MATH
, 1996
"... A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found ..."
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Cited by 24 (6 self)
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A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.
Lax Logical Relations
 In 27th Intl. Colloq. on Automata, Languages and Programming, volume 1853 of LNCS
, 2000
"... Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with th ..."
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Cited by 15 (2 self)
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Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the prelogical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.
Computing Vector Functions on Anonymous Networks
 SIROCCO '97. Proc. 4th International Colloquium on Structural Information and Communication Complexity
, 1997
"... We characterize exactly the vector functions f : # n # # n computable by several classes of anonymous networks. We study both undirectional and bidirectional networks, (partially) wireless networks, and two different processor activation models (synchronous and asynchronous). We also identify t ..."
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Cited by 15 (8 self)
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We characterize exactly the vector functions f : # n # # n computable by several classes of anonymous networks. We study both undirectional and bidirectional networks, (partially) wireless networks, and two different processor activation models (synchronous and asynchronous). We also identify the rigidity conditions under which a network can compute every function. Keywords: Anonymous networks, wireless networks, distributed computation, graphs, fibrations 1 Introduction Given a network, we can legitimately ask which functions can the network compute distributely. More precisely, given a domain of values # and a function f : # n # # n , where n is the number of processors of the network, we can ask whether the network (or any network out of a certain class) can compute f . The question is of course meaningful only if the network is anonymous, i.e., if the processors do not possess unique identifiers. Other problems, such as the election problem, have been studied on such ne...
Quasismooth Derived Manifolds
"... products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is ..."
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Cited by 14 (0 self)
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products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
A categorification of quantum sl(2
 Adv. Math
"... We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs betwee ..."
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Cited by 10 (4 self)
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We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs between 1morphisms are graded lifts of a semilinear form defined on ˙U. Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U˙ is constructed using iterated flag varieties that categorifies the irreducible (N + 1)dimensional representation of ˙ U.
Cofibrantly generated natural weak factorisation systems
, 2007
"... There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with r ..."
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Cited by 9 (0 self)
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There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C]. In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.’s. We then construct them by a method which is reminiscent of Quillen’s small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of mapswithstructure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on. 1
The basic geometry of Witt vectors
"... Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two di ..."
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Cited by 8 (2 self)
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Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two different ways. We allow not just ptypical Witt vectors but also, for example, those taken with respect to any set of primes in any ring of integers in any global field. We also allow not just padic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of the Greenberg transform. We investigate whether many standard geometric properties of spaces and maps are preserved by Witt vector functors.
Representing Nested Inductive Types Using Wtypes
"... We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive ..."
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Cited by 6 (3 self)
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We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive