
	Where results make sense

Topic: Bicategory

In the News (Fri 10 Jul 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Bicategory - Wikipedia, the free encyclopedia
		In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i.e.
		In a bicategory these 1-cells form a small category themselves by introducing mappings s, t, u,...
		The category of small categories, Cat, forms a bicategory with small categories as 0-cells, functors as 1-cells, and natural transformations as 2-cells.
	en.wikipedia.org /wiki/Bicategory (228 words)

	Bicategory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A bicategory is a concept in (Click link for more info and facts about category theory) category theory used to extend the notion of sameness (i.e.
		In a bicategory these 1-cells form a (Click link for more info and facts about small category) small category themselves by introducing ((genetics) the process of locating genes on a chromosome) mappings s, t, u,...
		The category of small categories, Cat, forms a bicategory with small categories as 0-cells, functors as 1-cells, and (Click link for more info and facts about natural transformation) natural transformations as 2-cells.
	www.absoluteastronomy.com /encyclopedia/b/bi/bicategory.htm (301 words)

	Paper on linear bicategories and non-commutative linear logic
		The structure of a bicategory may be viewed as the representation of tensor structure by composition (for example, functional composition, relational composition, etc.).
		Linear bicategories represent both the tensor and the par of linear logic as compositions.
		A number of examples of linear bicategories arising from different sources are described, and a number of constructions which result in linear bicategories are indicated.
	www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg00100.html (521 words)

	AMCA: Bicategories of quantales by Jan Paseka (Site not responding. Last check: 2007-10-08)
		The upshot is that known definition of Morita equivalence for this case amounts to isomorphism of objects in the pertinent bicategory.
		in (A, A) given by the canonical bimodule A --> A <-- A, is a bicategory [Q
		In the language of bicategories, two quantales are isomorphic objects in the bicategory [Q
	at.yorku.ca /c/a/f/o/11.htm (192 words)

	Search results au = (SEELY, R) (Site not responding. Last check: 2007-10-08)*
		Under the assumption that the domain bicategory is small and the codomain bicategory is locally cocomplete then this composite exists and a simple construction of it is given using local colimits.
		It is shown that in the module bicategory the module associated with this optransformation is right-adjoint to the module associated with the transformation.
		These are bicategories with two operations of bicategorical composition, tensor and cotensor, together with the additional requirement of linear distributivity of tensor over cotensor.
	www.math.mcgill.ca /~rags/rags_pubs.html (3446 words)

	categories: Re: Time for functors to grow up; three queries
		In the usual definition of bicategory, you have a binary composition of hom-categories and an identity cell on each object.
		The structure formed by bicategories is 3-dimensional: you have bicategories, weak functors, weak transformations, and modifications.
		So in order to state the equivalence between the two definitions of bicategories, you "ought" to have to make a statement involving 3-dimensional structures: specifically, that the two different tricategories of bicategories are triequivalent.
	north.ecc.edu /alsani/ct02(1-2)/msg00061.html (1040 words)

	Modules (Site not responding. Last check: 2007-10-08)
		The general notion of a module between two morphisms of bicategories is described.
		However, when the domain bicategory is small and the codomain bicategory is locally cocomplete then the composite of any two modules does exist and has a simple construction using the local colimits.
		In the module bicategory the module associated with this optransformation is right-adjoint to the module associated with the transformation.
	www.emis.de /journals/TAC/volumes/11/17/11-17abs.html (265 words)

	Monads and interpolads in bicategories (Site not responding. Last check: 2007-10-08)
		Given a bicategory, 2, with stable local coequalizers, we construct a bicategory of monads Y-mnd by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y. Any lax functor into Y factors through Y-mnd and the 1-cells turn out to be the familiar bimodules.
		The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchy-complete, but have a well-known Cauchy-completion in common.
		This prompts us to formulate a concept of Cauchy-completeness for bicategories that are not locally ordered and suggests a weakening of the notion of monad.
	138.73.27.39 /tac/volumes/1997/n8/3-08abs.html (253 words)

	Morphisms and modules for poly-bicategories (Site not responding. Last check: 2007-10-08)
		Linear bicategories are a generalization of ordinary bicategories in which there are two horizontal (1-cell) compositions corresponding to the ``tensor'' and ``par'' of linear logic.
		Benabou's notion of a morphism (lax 2-functor) of bicategories may be generalized to linear bicategories, where they are called linear functors.
		In this case we recover the notion of a linear bicategory.
	www.iti.cs.tu-bs.de /~koslowj/RESEARCH/poly.abstract.html (473 words)

	categories: Re: Time for functors to grow up; three queries"
		I want to thank Like Tom Leinster, I assume that David actually means to treat Cat as a bicategory rather than a 2-category; the rest of the posting seems to contain nothing about pseudofunctors, but rather about factorizations for ordinary functors.
		Here are a few references about factorization systems in 2-categories and bicategories: [1] Ross Street, Two-dimensional sheaf theory, JPAA 23(1982) 251-270 [2] Ross Street, Characterization of bicategories of stacks, SLN 962 [3] Carboni, Johnson, Street, and Verity, Modulated bicategories, JPAA 94(1994) 229-282.
		In [3], factorization systems (strong liberal, conservative) and (liberal, strong conservative) are described which may or may not exist on a bicategory.
	north.ecc.edu /alsani/ct02(1-2)/msg00065.html (652 words)

	Monoidal category - Wikipedia, the free encyclopedia
		It follows from these two conditions that any such diagram commutes: this is Mac Lane's "coherence theorem".
		A monoidal category may be regarded as a bicategory with one object.
		Many monoidal categories have additional structure such as braiding or symmetry: the references describe this in detail.
	en.wikipedia.org /wiki/Monoidal_category (395 words)

	Claudio Hermida
		This correspondence extends smoothly to one between bicategories and a localised version of representable multicategories.
		The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudo-S-algebras in terms of representability.
		Abstract: We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps.
	maggie.cs.queensu.ca /chermida (1304 words)

	Observational trees as models for concurrency
		We introduce morphisms of bicategories, more general than the original ones.
		Therefore these morphisms can be regarded as categories enriched in bicategories "on two sides".
		There is a composition of such enriched categories, leading to a simple kind of tricategory Caten whose objects are bicategories.
	www.dsi.uniroma1.it /~labella/CatArrpapers.txt (844 words)

	CMS Summer 2002 Meeting
		Recently, with the notions of ``poly bicategory'' and ``linear bicategory'', we generalized several bicategorical notions to a setting where there were two (implicit or explicit) ``horizontal'' compositions.
		Poly bicategories have poly 2-cells which may be regarded as arrows with strings of 1-cells as domains and as codomains; the representability of such arrows by ordinary arrows with single object domains and codomains gives the notion of linear bicategory.
		Usually constructing module (bi)categories requires some completeness and cocompleteness; in the present setting we can identify such conditions as necessary for the representability of the module poly bicategory, rather than for its existence.
	www.cms.math.ca /Events/summer02/abs/ct.html (3009 words)

	Minimal Realization in Bicategories of Automata - Rosebrugh, Sabadini, Walters (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: The context of this article is the program to develop monoidal bicategories with a feedback operation as an algebra of processes, with applications to concurrency theory.
		In this setting the automata are 1-cells in contrast with previous studies where they appeared as objects.
		As a consequence we are able to study the relation of minimization and minimal realization to serial composition of...
	citeseer.lcs.mit.edu /rosebrugh98minimal.html (530 words)

	Monads And Interpolads In Bicategories - Koslowski (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: Given a bicategory, Y, with stable local coequalizers, we construct a bicategory of monads mnd (Y) by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y.
		The essential associativity of both compositions follows in analogy to the case of modules in an ordinary bicategory, cf.
		J. Koslowski (1997) Monads and interpolads in bicategories.
	citeseer.ist.psu.edu /327558.html (520 words)

	Please title this page. (manifolds.html) (Site not responding. Last check: 2007-10-08)
		This approach to glueing structures is clearly related to Ehresmann's one, based on pseudogroups of transformations [Eh].
		On the other hand, our setting inscribes in Lawvere's remark [La] that interesting mathematical structures not only organize in categories, but are themselves categories, enriched over some suitable base: a monoidal category as in Lawvere's original formulation, or more generally a bicategory.
		The bases we actually use are suitable ordered categories (very particular bicategories).
	www.bangor.ac.uk /ma/news/manifolds.html (296 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Subject: categories: Re: Reading advise on bicategory theory john isbell writes: -The matter of coherence is wide open.
		This is shown for monoidal categories (bicategories with one object) e.g.
		The same argument works for arbitrary bicategories provided, in defining a multicategory, one replaces the free monoid generated by a set by the free category generated by a graph.
	www.mta.ca /~cat-dist/catlist/1999/bicat-reading (1339 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		In the purely algebraic setting, it is well known that rings are Morita equivalent if they are equivalent objects in a bicategory whose 1-cells are bimodules.
		We show that von Neumann algebras form a bicategory, having Connes's correspondences as 1-morphisms, and (bounded) intertwiners as 2-morphisms.
		Further, we prove that two von Neumann algebras are Morita equivalent iff they are equivalent objects in the bicategory.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20420433 (232 words)

	Re: quaternionic C*-algebras
		There's a very nice bicategory whose >objects are all the rings in the world.
		An algebraic benefit (?) of this is that the notion of a module or (especially) algebra over a commutative ring remains the same as the usual one, rather than being enlarged.
		Since R is a subring of all these rings, we can always require that scalar multiplication commute when the scalar is real.
	www.lns.cornell.edu /spr/2000-01/msg0021413.html (2568 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A general 'double adjunction', which appears often in concrete constructions, has a colax double functor left adjoint to a lax one.
		This cannot be viewed as an adjunction in some bicategory, because lax and colax morphisms do not compose well and do not form one.
		However, such adjunctions live within an interesting construct, the strict double category of weak double categories, with horizontal and vertical arrows provided by the lax and colax double functors, respectively, and a suitable notion of double cell.
	www.dima.unige.it /~grandis/Dbl.Adj.Abs.html (109 words)

	Workshop on Chu Spaces: Theory and Applications 25th June 2000.
		The cyclic Chu-construction for closed bicategories, generalizing the original Chu-construction for symmetric monoidal closed categories, turns out to have a non-cyclic counterpart.
		Both constructions are based on the so-called Chu-cell and can be generalized to chains of composable 1-cells.
		This leads to two hierarchies of closed bicategories for any closed bicategory B. Chu-cells in rel correspond to bipartite state transition systems.
	www2.parc.com /spl/members/paiva/chu-abs.html (1229 words)

	@CAT 2001-2002
		Abstract: The context is the program studying the bicategory of spans of graphs as an algebra of processes with applications to concurrency theory.
		This problem, which was addressed in the speaker's recent Ph D thesis, will be revisited with the help of the still more recent concept of enrichment in a bicategory.
		We will move on to a `horizontal' view of pro-W-cat and pro-W-pro (although as of this writing it would be disingenuous to say that this later work is `in press') and establish a biequivalence between the views introducing the audience to still further bicategorical constructs.
	www.mscs.dal.ca /~pare/Sem01-02.html (822 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		In a recent paper, R. Paré, D. Pronk, and I consider the construction by which adjoint arrows, and unit and counit 2-cells, are freely added to a category.
		It is easy to determine whether two objects or arrows in the resulting bicategory are equivalent, but we show that the equivalence of 2-cells is in general undecidable!
		The proof involves a simulation of an arbitrary 2-register abacus by the bicategory generated in this way from a suitable category.
	cs.stmarys.ca /~dawson/Abacus.html (608 words)

	@CAT 2002-2003
		We are in the process of showing that there is a coKZ-doctrine L on \K for which \lex = \K^L. With this at hand we will have a distributive law LR--->RL over \K giving \reg = \K^{RL} and completing the main intent of the project.
		Abstract: We shall study two different universal properties of the bicategory of spans.
		The free bicategory generated by an oplax one will be revealed for all to see.
	www.mscs.dal.ca /~pare/Sem02-03.html (1493 words)

	[No title]
		There's a bicategory whose objects are super algebras A, whose 1-cells M: A --> B are left A- right B- modules in V, and whose 2-cells are homomorphisms between modules.
		On the other hand, in the bicategory above we always have a biadjunction A tensor C --> D ------------------------- C --> A* tensor D (essentially because left A-modules are the same as right A-modules, where A denotes the super algebra opposite to A).
		Actually, one is an equivalence iff the other is, because both of these canonical 1-cells are given by the same A-bimodule, namely the one given by A acting on both sides of the underlying superspace of A (call it S) by multiplication.
	math.ucr.edu /home/baez/twf_ascii/week212 (3437 words)

	Representing Place/Transition Nets in Span(Graph) (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: The compact closed bicategory Span of spans of reflexive graphs is described and it is interpreted as an algebra for constructing specifications of concurrent systems.
		The Compact Closed Bicategory of Left Adjoints - Katis, Walters (1999)
		Bicategories of Processes - Katis, Sabadini, Walters (1997)
	citeseer.lcs.mit.edu /katis97representing.html (502 words)

	Re: quaternionic C*-algebras
		>> >>You may remember that a >>bicategory with one object is a monoidal category.
		>(All the associators etc exist because they existed in the smaller category, >and composition is defined when it should be, so this is a bicategory.) Umm, now I'm a bit nervous: in a monoidal category we can tensor morphisms as well as objects.
		It seems rather odd, using structure on your objects not to restrict the allowed morphisms, but merely to define the tensor product and internal hom in your category.
	www.lns.cornell.edu /spr/2000-01/msg0021421.html (1398 words)

Try your search on: Qwika (all wikis)