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Topic: Groupoid

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	PlanetMath: groupoid (Site not responding. Last check: 2007-11-07)
		The groupoid (or ``magma'') is closed under the operation.
		This is version 7 of groupoid, born on 2002-02-02, modified 2002-11-06.
		Since some groupoids of the first kind are also groupoids of the second kind, and vice versa, I think it would be beneficial to have two articles, one about groupoid (binary operation) and one about groupoid (category).
	planetmath.org /encyclopedia/Magma.html (133 words)

	Overview (Site not responding. Last check: 2007-11-07)
		In this example the groupoid appears as a refinement of the concept of moduli space, which parametrizes the objects to be classified, but in a way incorporating the symmetries of these objects as well.
		The theory of Lie groupoids is one approach to the problem of endowing an abstract groupoid with geometric structure.
		Groupoids were introduced by Brandt in 1926 in a paper on composition of quadratic forms in four variables.
	www.math.ubc.ca /~behrend/luminy/overview.htm (625 words)

	A very incomplete list of papers on groupoids
		Groupoids are found in the papers by Muhly, Qiu, and Solel; Peters; Ramsay and Walter; and Yamanouchi; as well as problems submitted by Paterson, Shultz, and Solel.
		Groupoids are used to reprove many of the theorems dealing with automorphisms and endomorphisms of a finitely generated free group.
		The main objective, as the title suggests, is to explore connections between the three concepts of groupoids, inverse semigroups and their operator algebras with a balanced approach between the details and the main ideas.
	www.cameron.edu /~koty/groupoids/1991.htm (2733 words)

	Magma (algebra) - Wikipedia, the free encyclopedia
		In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure.
		The term magma for this kind of structure was introduced by Bourbaki; however, the term groupoid is a very common alternative.
		Unfortunately, the term groupoid also refers to an entirely different kind of algebraic concept described at groupoid.
	en.wikipedia.org /wiki/Magma_(algebra) (535 words)

	A very incomplete list of papers on groupoids
		Mainly a study of foliations, this book includes two sections related to groupoids - one is an introduction to groupoids in the context of differential geometric structures and the other is on the classifying space for a topological groupoid.
		Renault, Jean, Two applications of the dual groupoid of a C*-algebra in Operator Algebras and their Connections with Topology and Ergodic Theory, Proceedings of the OATE Conference held in Busteni, Romania, Aug. 29-Sept. 9, 1983, Lecture Notes in Mathematics, 1132 (434-445), Springer-Verlag, New York, 1985.
		This book examines the topological groupoid of a foliation in the case of a foliated bundle with discrete structural group and the case of the Reeb foliation.
	www.cameron.edu /~koty/groupoids/1981.htm (1634 words)

	[No title]
		B is a Serre fibration then the fundamental groupoid ß1(E) has an addit* ional compatible groupoid* structure arising from the equivalence relation Eq(p) defined by the m* ap p; these two groupoid* structures define a double groupoid which we write fl(p), since it is defined b* *y methods of Galois theory.
		A definition of a homotopy d* ouble groupoid* of a pair of pointed spaces was made with P.J. Higgins in 1974, and exploited to o* btain a 2-dimensional Van Kampen type theorem for this double groupoid, and hence for Whitehead's cro *ssed module of a pair (see [4]).
		I is the fundamental groupoid * functor, and H the nerve functor; o y is the Yoneda embedding, r and i are the restrictions of R and I respect ively along y; explicitly, r is the singular simplex functor and i carries finite ordinals to codiscr ete groupoids* on the same sets of objects.
	hopf.math.purdue.edu /BrownR-Janelidze/dgpsmap5.txt (5652 words)

	Abstracts (Site not responding. Last check: 2007-11-07)
		A differential groupoid is a groupoid whose space of morphisms and space of units are differentiable manifolds (possibly with corners) such that all structural maps are differentiable and the domain map is a submersion.
		The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry.
		To a differential groupoid one can associate a Lie algebroid, which is a vector bundle whose sections identify with the right invariant vector fields on the groupoid that are tangent to the fibers of the domain map.
	www.math.psu.edu /nistor/abstracts.html (2189 words)

	[No title]
		the groupoids fibered over the groupoid of finite sets are included among kelly's objects, corresponding to operations on groupoids built out of basic tensor product (actually just the usual cartesian product of groupoids, but treated as just a tensor product not assumed to be cartesian) and 2-colimit operations (with distributivity of tensor product over 2-colimits).
		now we define the "homotopy cardinality" of a connected groupoid to be the reciprocal of the size of it's fundamental group, with additivity under discrete sums to extend the definition to arbitrary groupoids.
		the point of this concept of "homotopy cardinality" is to treat groupoids (or at least those of finite homotopy cardinality) as categorified positive real numbers, somewhat analogously to the way in which the ordinary concept of cardinality amounts to a way of treating finite sets as categorified natural numbers.
	math.ucr.edu /~jdolan/feynman (748 words)

	Atlases of Groupoids and Global Actions.
		Groupoid atlases are natural generalisations of A. Bak's global actions, which originated in his study of algebraic analogues of topological spaces.
		Such an object is a family of independant groupoids all having the same set of objects, so is the same as a multiple groupoid.
		In general a groupoid atlas is a set with a covering by patches, each part of which carries the structure of a single domain global atlas.
	www.informatics.bangor.ac.uk /~tporter/GAGA/agga.html (925 words)

	Determination Of A Double Lie Groupoid By Its Core Diagram - Brown, Mackenzie (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		...= TG the core is AG and the dual VB groupoid defines the cotangent groupoid structure.
		1 Fibrations and quotients of differentiable groupoids (context) - Higgins, Mackenzie - 1990
		The Monodromy Groupoid Of A Lie Groupoid - Brown, Mucuk (1996)
	citeseer.ist.psu.edu /brown92determination.html (607 words)

	Kirill C. H. Mackenzie, School of Mathematics & Statistics, University of Sheffield (Site not responding. Last check: 2007-11-07)
		Ordinary Lie groupoids consist of `arrows between points' and the elements of a double Lie groupoid can correspondingly be thought of as squares, the edges of which are arrows belonging to ordinary groupoid structures, and the vertices of which are points.
		My own work focuses on multiple Lie groupoid and multiple Lie algebroid structures and combinations of these, but there is a very large body of work on multiple category structures which can be said to have its origins in Ehresmann's work of the 1960s and which is today intimately bound up with work on quantization.
		Thus the iteration of the notion of groupoid is in a sense the most concrete example, apart perhaps from double vector bundles.
	www.shef.ac.uk /~pm1kchm/ms.html (511 words)

	Groupoid Fest Talks
		Under mild assumptions the C-algebra of a multigraph may be seen to be isomorphic to the C-algebra of a groupoid associated to the multigraph.
		For general G, the asymptotic morphism induces the analytic index map on the groupoid.
		Ramsay: Local action groupoids abstract: The title refers to groupoids with the property that every point in the space of units has a neighborhood to which the restriction of the groupoid is isomorphic to a groupoid arising from an action of a group on that neighborhood.
	math.la.asu.edu /~kaz/gfest98/gtalks.html (583 words)

	Abstract (Site not responding. Last check: 2007-11-07)
		Speaker: Alan Paterson, University of Missippi Title: Inverse semigroups, groupoids and their operator algebras Abstract: Inverse semigroups can be thought of as semigroups of partial one-to-one maps which are closed under inversion, or equivalently, as *-semigroups of partial isometries on a Hilbert space.
		A groupoid is roughly a set with a partially defined multiplication such that the usual group axioms hold whenever they make sense.
		The representation theories of the inverse semigroup and the groupoid are the same, and the well-developed representation theory of groupoids can thus be used to investigate the representation theory of inverse semigroups.
	www.math.unl.edu /~bharbour/colloqabs/paterson.html (197 words)

	Physics Help and Math Help - Physics Forums - LQG and Category Theory
		If over a set of n points you build a groupoid with arrows all the possible ordered pairs of points, then the canonical construction of an algebra over this groupoid is the one of n xn matrices.
		If you use the tangent groupoid to a manifold, the procedure of building an algebra is very similar to Weyl quantization.
		what makes a groupoid different from a group is that a group is a category with only one object, so all morphisms can be composed with all other morphisms.
	www.physicsforums.com /printthread.php?t=15179 (1471 words)

	Problems on groupoids (Site not responding. Last check: 2007-11-07)
		3) An equivalence of groupoids induces an isomorphism of cohomology.
		The construction of the tangent groupoid has been extended to local Lie groupoids in the context of pseudo-differential operators by Nistor, Weinstein and Xu.
		In particular, Cuntz-Krieger groupoids and AF groupoids are singly generated.
	unr.edu /homepage/ramazan/groupoid/open_prb/gop.html (629 words)

	"Foliation groupoids and their cyclic homology" (Site not responding. Last check: 2007-11-07)
		A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective.
		Moreover, we show that among the Lie groupoids integrating a given foliation, the holonomy and the monodromy groupoids are extreme examples.
		The second theorem shows that the cyclic homology of convolution algebras of foliation groupoids is invariant under Morita equivalence of groupoids, and we give explicit formulas.
	www.math.uu.nl /people/crainic/absflgrnou.html (203 words)

	Atlas: $C^{\ast }$-algebras associated to groupoids with proper orbit space by Madalina Buneci (Site not responding. Last check: 2007-11-07)
		-algebra of a locally compact groupoid was introduced by Jean Renault in [2]: the space of continuous functions with compact support on groupoid is made into a * -algebra and endowed with the smallest C
		For defining the convolution on a locally compact groupoid, one needs an analogue of Haar measure on locally compact groups.
		In the case of a transitive groupoid the C
	atlas-conferences.com /cgi-bin/abstract/cane-86 (238 words)

	Frequently asked questions (Site not responding. Last check: 2007-11-07)
		A groupoid (sometimes also called binary system) is a nonempty set with one binary operation.
		Groupoid is a small category in which every morphism is an isomorphism.
		Nevertheless, there are hundreds, perhaps thousands of papers on groupoids in the sense of binary systems, while there are only tens of papers on groupoids meant as these structures between groups and categories.
	www.karlin.mff.cuni.cz /utf/~jezek/FAQ-G.htm (771 words)

	The Holonomy Groupoid Of A Locally (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Topological Groupoid Mohammed A. Aof Department of Mathematics...
		Abstract: The well known holonomy groupoid of a foliation is here generalised to the holonomy groupoid of a locally topological groupoid.
		4 The monodromy groupoid of a Lie groupoid - Brown, Mucuk - 1995
	citeseer.ist.psu.edu /607814.html (243 words)

	A homotopy double groupoid of a Hausdorff space (Site not responding. Last check: 2007-11-07)
		We associate to a Hausdorff space, $ X $, a double groupoid, $ \mbox{\boldmath $ \rho $}^{\square}_{2} (X) $, the homotopy double groupoid of $ X $.
		The construction is based on the geometric notion of thin square.
		Under the equivalence of categories between small $ 2 $-categories and double categories with connection the homotopy double groupoid corresponds to the homotopy 2- groupoid, $ {\bf G}_{2} (X) $.
	www.tac.mta.ca /tac/volumes/10/2/10-02abs.html (165 words)

	[No title]
		If the empty string is returned, no name has been given."; Groupoid::usage = "Groupoid is the 'head' for a pair with the first component a set of elements and the second an operation.
		Groupoids are the basic structure of this package.
		GroupoidQ[{G1,G2,..}] acts on a list and returns a set of Boolean values."; GroupoidQ::usage = "GroupoidQ[G] is a Boolean function, returning True if G consists of a pair with the first component a set of elements and the second an operation.
	archives.math.utk.edu /software/multi-platform/mathematica/ntbks/abstractalg/ExploreAbsAlgMath/AbbrevGrpPck.ma (3776 words)

	AMCA: Groupoids orthogonal to quasigroups and their application by Smile Markovski (Site not responding. Last check: 2007-11-07)
		We focus on finite quasigroups satisfying identity x(xy)=y or x(xy)=xy or x(xy)=yx and symmetric ones, and then their orthogonal complement is a left-zero groupoid or a quasigroup, respectively.
		In many cases we have seen that if a quasigroup posesses an orthogonal complement which is left(right) zero groupoid or quasigroup itself then the so called quasigroup string processing gives as a result "fractal" image.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/e/e/58.htm (188 words)

	AMCA: Developability and nonpositive curvature by Andre Haefliger (Site not responding. Last check: 2007-11-07)
		In the first lecture, which is meant to be elementary, we shall consider the particular case of orbifolds to illustrate the notions of coverings and developability, notions extended later on to the more general case of étale groupoids.
		Such groupoids arise naturally in the theory of Riemannian foliations and in the theory of complexes of groups.
		This means that such a groupoid is equivalent to the groupoid associated to an action of a group G on a complete simply-connected metric space [X\tilde] of non-positive curvature.
	at.yorku.ca /c/a/b/b/31.htm (279 words)

	Physics Help and Math Help - Physics Forums - View Single Post - LQG and Category Theory
		If over a set of n points you build a groupoid with arrows all the possible ordered pairs of points, then the canonical construction of an algebra over this groupoid is the one of nxn matrices.
		"why are the arrows of a groupoid not forming a group?"
		a groupoid do not form a group unless it could be that
	www.physicsforums.com /showpost.php?p=154784&postcount=5 (256 words)

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