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Topic: Enriched category

	Springer Online Reference Works
		It follows from the definition that each object in a category has a unique identity morphism; thus it is possible to identify objects with their identity morphisms, leading to an axiomatization of categories in which "morphism" and "composite" are the only primitive notions (see [9]).
		They are distinguished from the general run of enriched categories by the possibility of considering diagrams within them which commute "up to isomorphism" but not exactly; the weaker notion of a bicategory [a17] is a further expression of this idea.
		In these areas of category theory coherence theorems play an important part: these are theorems which allow one to deduce the commutativity of a large class of diagrams from that of certain particular diagrams (see [a19], for example).
	eom.springer.de /c/c020740.htm (2903 words)

	Enriched category - Wikipedia, the free encyclopedia
		In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.
		An enriched functor is the appropriate generalization of the notion of a functor to enriched categories.
		Enriched functors are then maps between enriched categories which respect the enriched structure.
	en.wikipedia.org /wiki/Enriched_category (806 words)

	[No title]
		The ar* gu- ments of [9] are given within the framework of a model structure for presheaves of groupoids enriched* in simplicial sets whose associated homotopy category is equivalent to the standard homotopy category of simplicial presheaves.
		A cofibration in the category of enriched A-diagrams is a map which has the left lifting property with respect to all trivial fibrations.
		Ob (A)________//BA is a homotopy cartesian diagram of bisimplicial presheaves.
	www.math.purdue.edu /research/atopology/Jardine/diagrams.txt (6047 words)

	BioMed Central \| Full text \| GOTree Machine (GOTM): a web-based platform for interpreting sets of interesting genes ...
		In order to map the granular annotations such as "apoptosis" to general categories like "cell death", GO files for the 3 main categories were downloaded from the current ontologies section from the Gene Ontology consortium website [17] as flat text files and parsed by a Perl script.
		Enriched GO categories are colored red, and genes in each category are also included in the exported GOTree (for an example, see http://genereg.ornl.gov/gotm/paper/testis_output.html).
		The GO categories, homeostasis and blood coagulation, are also enriched, which may be consistent with the ability of liver to synthesize various protein molecules that are responsible for clotting of blood.
	www.biomedcentral.com /1471-2105/5/16 (4769 words)

	Kestrel Institute - Research Staff - Dusko Pavlovic - Semantics of computation
		In the present paper, we describe a category of processes modulo strong bisimulations, with the bisimilarity preserving simulations as morphisms, and show that it is equivalent to the category of labelled irredundant trees and the label preserving tree morphisms.
		An abstract construction of a category of processes in a general setting is presented in the appendix.
		We further discuss categories of resumptions and of hyperfunctions, which are the main examples of prcess categories.
	www.kestrel.edu /home/people/pavlovic/semantics.html (1844 words)

	ACG - Homepage (Site not responding. Last check: 2007-10-25)
		Enriched category theory (ECT) intends to described richer interactions of objects.
		Roughly speaking, interactions between objects a,b of a (classical) category are captured by the set, Hom(a,b) of arrows from a to b.
		In an enriched category the Hom(a,b) may not be a set anymore but roughly any other good kind of object.
	www.cwi.nl /htbin/acg/ACG.php4?tid=51 (162 words)

	USA Rice Federation: Industry: News
		While the industry supports the goal of increasing whole grain consumption, enriched grains, which are a key source of the B vitamins including folate, play a major role in healthy eating and disease prevention and should continue to be emphasized.
		Enriched and whole grains should be depicted together in the grain category.
		Separating enriched from whole grains in a new graphic would be confusing to consumers and could imply that enriched grains are not recommended.
	www.usarice.com /news/news_detail.cgi/76/5 (391 words)

	The Dimensional Ladder
		Categories of Mathematical Objects Definition of category There are many categories of mathematical gadgets, but we'll consider three: Set, Vect and Top, the latter two because they don't arise very quickly from category theory itself.
		Categories from Spaces The fundamental groupoid of a topological space The fundamental group of a pointed space categories from chain complexes: a 2-term chain complex is a category in AbGp.
		Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	enriched Sellers, Suppliers, Exporters, Manufacturers, Factories (Site not responding. Last check: 2007-10-25)
		FEBICO® Selenium Enriched Spirulina is a highly nutritious blue-green algae and a rich source of vitamins, Beta carotene, minerals, chlorophyll, gamma-linolenic acid (GLA) and protein.
		PURELAC is a milk powder enriched with nutrients, similar to half-skimmed milk.
		MILKMAX is a milk powder enriched with nutrients, available in instant version, enriched with calcium, developed especially for human necessities (light, easy to digest, indicated to kids and the elderly).
	www.fuzing.com /qrx/enriched (484 words)

	[No title]
		.,.An) is a trivial category with one object and one morphism.
		The formulation of the category of symmetric spectra that follows is therefore * a simple generalization of the category* of symmetric spectra of [9].
		Ring Categories, Bipermutative Categories, and the Operads * and E * This section is devoted to the proofs of Theorems 3.4 and 3.8.
	hopf.math.purdue.edu /Elmendorf-Mandell/RMA2.txt (17071 words)

	[No title]
		A model category is called stable if the initial and terminal objects coincide (that is, it is a pointed category) and if the induced suspension functor is in* vertible on the homotopy category*.
		The category UadC is an sAb-model category, and therefore Sp (UadC=S) is an Sp (sAb)-model category.
		A D-model category is a model category M which is also a closed D-module category and where the two conditions from [Ho2, 4.2.18] hold: these are again the analog of SM7 and a un* *it condition.
	hopf.math.purdue.edu /Dugger/spenrich.txt (12833 words)

	York University: Category seminar
		Under certain hypotheses, this simple remark may be extended to lax algebras, and leads by way of the Kleisli category of the associated monad to a certain "neighborhood presentation" of the theory of lax algebras.
		ABSTRACT: Following the description by Manes [1] of the category of compact Hausdorff spaces as the Eilenberg-Moore category for the ultrafilter monad, Barr [2] showed that by weakening the axioms for a monad and the subsequent algebras, the Eilenberg-Moore category could be seen to be isomorphic to the category of topological spaces.
		Surprisingly, their braided monoidal categories have played a starring role in the recent resurgence of interest n knot theory led by the work of Vaughan Jones.
	www.math.yorku.ca /Seminars/category (2299 words)

	Ars Mathematica » Blog Archive » Opinions of Category Theory
		Category theory does not help me at all when I am trying to figure out the long term behaviour of some function given by an ODE - so I don’t use it when I am doing that.
		Your group theory comment is a straw man. category theory is not the natural domain of discourse of the Sylow theorems (or, never underestimate the power of a theorem that counts something).
		While category theory is the natural language of sets with structure, it doesn’t usually capture the “thingness” of any particular class of objects, the quality that makes those objects what they are (I probably sound like Heiddeger now).
	www.arsmathematica.net /archives/2006/06/24/opinions-of-category-theory (3632 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		Category theory, acyclic models and its applications in homological algebra.
		Category theory, higher dimensional algebra, holonomy, groupoids and crossed objects in algebraic topology.
		Category theory and its applications to algebra, topology and computer science.
	jhrs.rmi.acnet.ge /interests.htm (227 words)

	The University of Sydney - Categories and Combinatorics
		The University of Sydney - Categories and Combinatorics
		Category theory is a recent branch of mathematics originating in algebraic topology, but rapidly establishing connections with algebra, logic, algebraic and differential geometry, and most recently computer science.
		The main interests of the Combinatorics group concern applications of category theory to computer science, one of the hottest research areas of theoretical computer science, and to algebraic combinatorics, the name given to the recent cross-fertilization between two great areas of mathematics: algebra and combinatorics.
	www.maths.usyd.edu.au /res/Catecomb.html (199 words)

	WADT'99: 36 (Kinoshita) (Site not responding. Last check: 2007-10-25)
		Moreover, we give a new definition of data refinement, using the notion of enriched adjunction, and show that upward simulations are complete with respect to the new definition of data refinement.
		Our definition of data refinement is not chosen in an ad hoc manner, but it is given as a part of enriched adjunction, which is a natural and essential notion in category theory.
		Although the category theory we use is not old (less than 10 years!), I explicitly claim that this is not a work in category theory, but a work in computer science which uses category theory.
	www-lsr.imag.fr /WADT99/Abstracts/36.html (395 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		The second category is HEU declared excess to national security requirements in the United States and committed to peaceful purposes.
		It provides a rigorous and comprehensive assessment of the amounts of plutonium and HEU in military and civilian programmes, in nuclear and non-nuclear weapon states, and in countries seeking to acquire nuclear weapons.
		Low-enriched uranium (with enrichment levels less than 20 percent) was produced until 1985, when the plant was placed in a standby mode because of declining demand for low-enriched uranium from the commercial nuclear power sector; the plant was permanently closed in 1987.
	lycos.cs.cmu.edu /info/enriched-uranium--highly-enriched-uranium.html (630 words)

	Enriched category (Site not responding. Last check: 2007-10-25)
		Emami is test marketing a new variant â” âSona Chandi Chocoprashâ enriched with chocolate for children.
		We will define what it means for C to be an enriched category over the monoidal category M.
		In each of the examples above, there is such a functor from M to the category of sets, so each kind of enriched category in the examples can also be described as an ordinary category with certain additional structure or properties.
	publicliterature.org /en/wikipedia/e/en/enriched_category.html (599 words)

	2-category - Wikipedia, the free encyclopedia
		In category theory, a 2-category is a category with "morphisms between morphisms".
		It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure induced by the composition).
		For all objects A and B, a category
	en.wikipedia.org /wiki/2-category (172 words)

	Ringoids \| The n-Category Café
		The idea is to fix a category X and define a X-category to have a set of objects and, for any pair of objects a and b, not a set but an object of X called hom(a,b).
		We can write down the whole definition of category this way as long as X is a monoidal category, that is, a category with tensor product.
		Kelly was the first to bring category theory to Australia, and enriched category theory has been a mainstay of Australian category theory ever since he invented it.
	golem.ph.utexas.edu /category/2006/09/ringoids.html (3031 words)

	Categories & Products
		All categories at the same level in catalogue tree are sorted by priority.
		Categories with the same priority are sorted alphabetically.
		Category Image: Click Browse and navigate to the image that you would like to upload for the category.
	www.pinnaclecart.com /member/help/v3.10/Categories_Products.htm (2247 words)

	A Neighborhood of Infinity: Monads for vector spaces, probability and quantum mechanics pt. I
		Enriched categories are powerful stuff, and I should really learn more about them.
		Enriching in Ab gives an abelian group structure on your hom-sets, so you can add morphisms; enriching in Top gives a topological structure to your hom-sets.
		There's a rich and detailed theory of enriched categories, but the short version is apparently "if E is symmetric monoidal closed, then all of ordinary category theory carries over, mutas mutandis".
	sigfpe.blogspot.com /2007/02/monads-for-vector-spaces-probability.html (1731 words)

	Casual Category Theory - Spring 2000
		A V-enriched category, in few words, is a category having hom-objects in V, where V is a monoidal category.
		Proofs of some well-known properties of presheave categories, namely: presheave categories are free colimit completions or any presheave functor is isomorphic to a colimit of representables.
		The idea is that as such, some of them will be preserved by adjoints relating T categories of other models (synchronisation trees, event structures, petri nets,...).
	www.brics.dk /~varacca/CCT/cct-spring00.html (504 words)

	categorified associahedra (Site not responding. Last check: 2007-10-25)
		A familiar example of this process is the definition of an enriched category where V is monoidal and the usual morphism sets are replaced by morphism objects.
		Let's try it with the definition of bicategory, where there are usually categories of morphisms.
		The unit conditions for the enriched bicategory are not hard to draw as well.
	faculty.tnstate.edu /sforcey/ct06.htm (564 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		Especially, the "extensibility" phenomenon Hoare mentioned about refinement in his unpublished draft is explained by enriched adjunction and we can say that such extension is possible whenever the language construct can be described as algebraic structure on LocOrd_l, defined in the next paragraph.
		The category of locally ordered categories and locally ordered functors has monoidal structure given by so called Gray tensor.
		Moreover, that category is enriched over itself, by considering the category of locally ordered functors and lax transformations.
	www.math.mcgill.ca /rags/seminar/yoshiki.txt (356 words)

	Further Work
		of sign systems with semiotic morphisms, which is actually an "order enriched category," because it is enriched with a priority ordering on the morphisms.
		Category theory suggests that the right definition of blend is characterized by the "optimality" or "universal" property of "pushouts" (which must be in some sense "lax" because of the ordering); see [B] for details.
		Empirical work will be useful for learning more about the orderings on representations, thus providing better foundations for measuring the adequacy of a given representation for a given task.
	www-cse.ucsd.edu /users/goguen/papers/sm/node7.html (747 words)

	Isar Stubbe
		November 12, 2003 :: ''Docteur en mathématiques'' (PhD) with dissertation entitled ''Categorical structures enriched in a quantaloid: categories and semicategories'', supervisor: F. Borceux, Université Catholique de Louvain, Belgium.
		August 1999 :: Talk :: ''Why the category of sup-lattices may be of interest to physicists'', Australian category theory seminar, Sydney, Australia.
		Category Theory Symposium as satellite to the International workshop on current research in operational quantum logic, Brussels, Belgium.
	www.win.ua.ac.be /~istubbe (2020 words)

	USDA Reports
		grain consumption, enriched grains, which are a key source of the B vitamins
		Enriched and whole grains should be depicted together in the
		Separating enriched from whole grains in a new graphic
	www.riceonline.com /USARice082504.html (411 words)

	[No title]
		Date: Sat, 16 Sep 2006 10:19:01 +0100 Regarding enriched category theory, there is Max Kelly's book "basic concepts of enriched category theory".
		This is mainly about the theory of enriched categories based on a symmetric monoidal closed category.
		Now if one considers the SSet-category [A,S] where A is a SSet-category and S is the SSet-category SSet enriched over itself then it happens that [A,S] is an SSet-category with all the SSet-indexed limits and colimits and thus it has in particular the cotensors and tensors as defined by Kelly and al.
	www.lehigh.edu /~dmd1/vs918 (590 words)

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