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Topic: Presheaf

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	Presheaf Models for Concurrency (Site not responding. Last check: 2007-10-20)
		Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a built-in notion of bisimulation.
		We show how presheaf categories, in which traditional models of concurrency are embedded, can be used to deduce congruence properties of bisimulation for the traditional models.
		Presheaf models can be obtained as solutions to recursive domain equations.
	www.daimi.au.dk /~luca/thesis.html (389 words)

	presheaf (Site not responding. Last check: 2007-10-20)
		The first is the concept of presheaf, which formalizes the idea of restriction, and can be formulated in terms of elementary category theory.
		In the language of category theory, all of this can be summarized as follows: a presheaf of C on X is a contravariant functor from the category of open subsets of X, with inclusions as morphisms, to C.
		A sheaf is a presheaf satisfying an additional axiom which captures the idea of pasting together the structures F(U).
	www.yourencyclopedia.net /Presheaf.html (1996 words)

	[No title]
		In the case when A is a presheaf of groupoids G, this assignment of homotopy colimits determines an equivalence of homotopy categories Ho(s Pre(C=G)) ' Ho(s Pre(C)=BGop) (1) which generalizes the known relationship [3] between diagrams of simplicial sets defined on a groupoid H and that of simplicial sets over BH.
		Suppose that X is a presheaf on C. The model structure on the category s Pre(C)=X which arises from the topology on C=X is induced from the model structure on the category s Pre(C) of simplicial presheaves.
		There is a presheaf Morn(Aop) which consists of strings of arrows of length n in the presheaf of categories Aop, and holim---!AopX is the diagonal of a bisim* *plicial sheaf which is given by the object X xOb(A)Mor nAop in horizontal degree n.
	hopf.math.purdue.edu /Jardine/stack-coh6.txt (11151 words)

	PlanetMath: presheaf
		A presheaf with values in the category of sets (or abelian groups) is called a presheaf of sets (or abelian groups).
		Then a presheaf is merely a contravariant functor
		This is version 2 of presheaf, born on 2001-12-20, modified 2002-08-24.
	planetmath.org /encyclopedia/Presheaf.html (116 words)

	Citations: Presheaf models for concurrency - Cattani, Winskel (ResearchIndex)
		Presheaf models have been shown to include traditional models like synchronisation trees and event structures [13] along with their notion of bisimulation, to be related by powerful preservation properties associated with colimit preserving functors
		The presheaf models so obtained coincide with those of Section 4, whilst the relations will be shown to be in accordance with open map bisimulation (viz.
		that any colimit preserving functor between presheaf categories preserves bisimulation, which besides obvious uses in relating semantics in different models with different notions of bisimulation is, along with several other general results, useful in establishing congruence properties of process.
	citeseer.ist.psu.edu /cs?q=dbnum=1,GID=491353,DID=60148,start=50,cluster=none,qtype=context: (1556 words)

	PlanetMath: sheaf (Site not responding. Last check: 2007-10-20)
		The resulting presheaf is called, for obvious reasons, the restriction presheaf of
		Example 6 For an example of a presheaf that is not a sheaf, consider the presheaf
		presheaf, section, morphism of sheaves, isomorphism of sheaves, sheaf isomorphism
	planetmath.org /encyclopedia/SheafIsomorphism.html (708 words)

	Logic and Semantics Seminar - 10th October, 1997: Luca Cattani
		Presheaf categories has been proposed as abstract models of concurrency, with an inbuilt notion of bisimulation based on open maps.
		The Yoneda Lemma then justifies an intuition that a presheaf can be thought of as specifying for a typical path object the set of computation paths of shape that object.
		With this intuition presheaf models encompassing more traditional ones can be derived from natural definitions of path categories expressed as dense and full subcategories of the traditional models.
	www.cl.cam.ac.uk /Research/LS/Talks/1997_98/97_10_10.Abstract.html (413 words)

	[No title]
		Here, A[-n] means the presheaf of chain complexes which consists of a copy of A concentrated in degree_n, and then we obtain K(A, n) by applying the Eilenberg-Mac Lane W construction to obtain a simplicial abelian group in each section.
		The simplicial presheaf X x 1 is a cylinder object for a simpl* i- cial presheaf* X, so we can assume that the homotopy equivalence is simplicial.
		The presheaf of spectra GK=`(1=fi) is globally fibrant with respect to the Nisnevich topology, since direct image functors preserve global fibrations (this is a ubiquitous fact, which first appeared in [15]).
	hopf.math.purdue.edu /Jardine/gen-shea.txt (8175 words)

	Sheaf (mathematics) - Wikipedia, the free encyclopedia
		A presheaf satisfying only the uniqueness part of the sheaf axiom is sometimes called a monopresheaf.
		In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples.
		1955 Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
	en.wikipedia.org /wiki/Sheaf_space (2920 words)

	Towards an Operational Understanding of Presheaf Models - Nygaard (ResearchIndex)
		Elements of presheaves are understood as SOS derivations and as configurations of event structures.
		With their built in notion of open-map bisimulation, presheaf models have been put forward as providing a domain theory for concurrency.
		2 A fully abstract presheaf semantics for SCCS with finite del..
	citeseer.ist.psu.edu /nygaard01towards.html (612 words)

	[No title]
		Fix a topological space X. A presheaf on X is just a contravariant functor (from the category on open subsets of X, where morphisms are inclusions, to some other category).
		To be precise: the presheaf of nilpotent elements of the structure sheaf isn't always a sheaf.
		Because of this stalk issue, and because the stalk of a presheaf is the stalk of the associated sheaf, psi^{-1} is an exact functor from the category of sheaves on Y to the category of sheaves on X. +++ 3.8 Sheaves of pseudo-discrete spaces.
	math.stanford.edu /~vakil/ega0 (5238 words)

	Sheaf Article, Sheaf Information (Site not responding. Last check: 2007-10-20)
		A presheaf is similar to a sheaf, but it may not bepossible to glue.
		The first step is to introduce the concept of a presheaf, whichcaptures the idea of associating local information to a topological space.
		Suppose X is a topological space, and C is a category (often, this is the category of sets, the category of Abelian groups, the category of commutativerings, or the category of modules over a fixed ring).
	www.anoca.org /open/sheaves/sheaf.html (2552 words)

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