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33
Relations in Concurrency
"... The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the seman ..."
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Cited by 273 (34 self)
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The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higherorder processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higherorder processes. For this it is useful to generalise event structures to allow events which “persist.”
Higher rank graph C*algebras
, 2000
"... Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the ..."
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Cited by 58 (11 self)
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Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C ∗ –algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from “commuting” rank 1 graphs is given.
Tamagawa Numbers for Motives with (NonCommutative) Coefficients
 DOCUMENTA MATH.
, 2001
"... Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes a ..."
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Cited by 45 (16 self)
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Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, PerrinRiou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective Astructure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is nonmaximal. In each such case the conjecture with respect to a nonmaximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.
Generalized FourierMukai transforms
, 1996
"... Abstract. We study a generalization of the FourierMukai transform for smooth projective varieties. We find conditions under which the transform satisfies an inversion theorem. This is done by considering a series of four conditions on such transforms which increasingly constrain them. We show that ..."
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Cited by 20 (2 self)
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Abstract. We study a generalization of the FourierMukai transform for smooth projective varieties. We find conditions under which the transform satisfies an inversion theorem. This is done by considering a series of four conditions on such transforms which increasingly constrain them. We show that a necessary condition for the existence of such transforms is that the first Chern classes must vanish and the dimensions of the varieties must be equal. We introduce the notion of biuniversal sheaves. Some examples are discussed and new applications are given, for example, to prove that on polarised abelian varieties, each Hilbert scheme of points arises as a component of the moduli space of simple bundles. The transforms are used to prove the existence of numerical constraints on the Chern classes of stable bundles.
Theory and Applications of Crossed Complexes
, 1993
"... ... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for t ..."
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Cited by 17 (2 self)
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... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the rcube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the
Stable topological cyclic homology is topological Hochschild homology
, 1992
"... 1.1. Topological cyclic homology is the codomain of the cyclotomic trace from algebraic Ktheory trc:K(L) → TC(L). It was defined in [2] but for our purpose the exposition in [6] is more convenient. The cyclotomic trace is conjectured to induce a homotopy equivalence after pcompletion for a certain ..."
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Cited by 7 (5 self)
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1.1. Topological cyclic homology is the codomain of the cyclotomic trace from algebraic Ktheory trc:K(L) → TC(L). It was defined in [2] but for our purpose the exposition in [6] is more convenient. The cyclotomic trace is conjectured to induce a homotopy equivalence after pcompletion for a certain class of rings including
What are iteration theories
, 2007
"... “In the setting of algebraic theories enriched with an external fixedpoint operation, the notion of an iteration theory seems to axiomatize the equational properties of all the computationally interesting structures of this kind.” S. L. Bloom and Z. ´Esik (1996), see [3] We prove that iteration t ..."
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Cited by 7 (5 self)
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“In the setting of algebraic theories enriched with an external fixedpoint operation, the notion of an iteration theory seems to axiomatize the equational properties of all the computationally interesting structures of this kind.” S. L. Bloom and Z. ´Esik (1996), see [3] We prove that iteration theories can be introduced as algebras for the monad on the category of signatures assigning to every signature the rational tree signature. This supports the claim that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: is the monad of free rational theories and every rational theory has a continuous completion. 1.
The geometry of points on quantum projectivizations
 J. Algebra
"... Noncommutative projective bundle ..."
ADDING INVERSES TO DIAGRAMS ENCODING ALGEBRAIC STRUCTURES
, 2008
"... Abstract. We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal’s mod ..."
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Cited by 4 (3 self)
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Abstract. We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal’s model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups. 1.
Simplicial monoids and Segal categories
 Contemp. Math
"... Abstract. Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. ..."
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Cited by 4 (3 self)
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Abstract. Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids.