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Topic: Category of preordered sets

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Preorder

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	Preorder Encyclopedia Article @ Preferably.org (Site not responding. Last check: 2007-10-26)
		A set that is equipped with a preorder is called a preordered set.
		If a preorder is also equivalence relation, that is, a ≤ b and b ≤ a implies a = b, then it is a Glossaries.
		Then the quotient set, S / ~, is the set of all transitive closure of ~.
	www.preferably.org /encyclopedia/Preorder (605 words)

	PlanetMath: endomorphism
		In the most general case endomorphisms are encountered in category theory.
		The sets of endomorphisms and automorphisms for an object
		Endomorphisms also can be considered as objects of category of intermorphisms and (if the set of morphisms of our category is preordered) also of category of pseudomorphisms.
	planetmath.org /encyclopedia/Endomorphism2.html (185 words)

	Enriched category
		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.
		If M is the category of small sets[?], then C is a locally small category, because the hom-sets will all be small.
		If M is the category 2 with Ob(2) = {0,1}, a single nonidentity morphism (from 0 to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set.
	www.ebroadcast.com.au /lookup/encyclopedia/en/Enriched_category.html (560 words)

	Category Theory (Stanford Encyclopedia of Philosophy)
		Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.
		Category theory is an alternative to set theory as a foundation for mathematics.
		The very definition of a category is not without philosophical importance, since one of the objections to category theory as a foundational framework is the claim that since categories are defined as sets, category theory cannot provide a philosophically enlightening foundation for mathematics.
	plato.stanford.edu /entries/category-theory (11769 words)

	Field of sets (Site not responding. Last check: 2007-10-26)
		Given a topological space the clopen sets trivially form a topological Field of sets as each clopen set is its own interior and closure.
		The preorder Fields obtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold.
		Given an interior algebra, by replacing the topology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field.
	field-of-sets.mindbit.com (1444 words)

	Category of preordered sets biography .ms (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		The category Ord has preordered sets as objects and monotonic functions as morphisms.
		This is a category because the composition of two monotonic functions preserves monotonicity.
		The empty set (considered as a preordered set) is the initial object of Ord; any singleton preordered set is a terminal object.
	www.biography.ms.cob-web.org:8888 /Category_of_preordered_sets.html (123 words)

	Category theory (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		Category is called locally small iff for every two objects A and B there is a set Mor(''A'',''B'') of morphisms from A to B.
		Main articles universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.
		A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets).
	category-theory.kiwiki.homeip.net.cob-web.org:8888 (2701 words)

	Preorder at AllExperts (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		For preorder traversal of a tree data structure, see tree traversal.
		If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order, on the other hand, if it is also symmetric, that is, a ≤ b implies b ≤ a, then it is an equivalence relation.
		A preorder which is preserved in all contexts is called a precongruence.A precongruence which is also symmetric (i.e.
	experts.about.com.cob-web.org:8888 /e/p/pr/Preorder.htm (407 words)

	Lambek and Scott: Introduction to higher order categorical logic (Site not responding. Last check: 2007-10-26)
		Concrete categories are structured sets and functions preserving the structure (we must, of course, have identities and compositions).
		Examples: Sets are discrete categories, monoids are categories with only one object, preorders are categories with each Hom set of cardinality at most 1.
		When we consider sets, monoids, and preorders as categories, functors correspond to the appropriate notion of morphism.
	www.andrew.cmu.edu /user/cebrown/notes/lambekscott.html (4587 words)

	Chu Spaces Live (Site not responding. Last check: 2007-10-26)
		Notice that when A and B are set to the same space there is only one endomorphism, the identity, and when they are set to any two distinct spaces there are no maps between the two spaces at all.
		That is, each object c of C with underlying set U(c) can be represented as a Chu space having U(c) for its set of points such that the continuous functions between any two representing Chu spaces are exactly the functions (morphisms f:c->c' made concrete as U(f):U(c)->U(c')) between the represented objects c and c'.
		As a structurally attractive variant of this embedding, a symmetric "bi-Yoneda" embedding is possible for the special case where the concreteness of the embedded category is derived from its arrows by taking the elements of an object to be the set of arrows to that object.
	boole.stanford.edu /live (5661 words)

	Preorder - Wikipedia, the free encyclopedia
		If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.
		Then the quotient set, S / ~, is the set of all equivalence classes of ~.
		A net is a directed preorder, that is, each pair of elements has an upper bound.
	en.wikipedia.org /wiki/Preorder (481 words)

	[No title]
		Duality is of necessity between categories, and involves associating an object (say an algebra or space) of one category with its dual in another, or in the same category in the self-dual case.
		This new category of graphs is not the old one as its objects now have vertices in their own right, but it is equivalent to the old one.
		Or, the category of representaions of a quiver.
	www.mta.ca /~cat-dist/archive/2006/06-5 (15280 words)

	ONT Higher Order Categorical Logic
		A 'concrete category' is a collection of two kinds
		arbitrary sets and its morphisms are arbitrary mappings.
		Its objects are preordered sets, that is, sets
	suo.ieee.org /ontology/msg03373.html (274 words)

	ONT Re: Higher Order Categorical Logic -- Discussion (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		for a concrete category, the objects are sets.
		for a concrete category, the arrows are mappings between sets.
		(this is a case of trivial structure to preserve.) the next examples are sets plus "structure", in these cases, something like a "sums table", a "times table", or a "less than" relation is defined on the sets of the category and also preserved by the arrows of the category.
	suo.ieee.org.cob-web.org:8888 /ontology/msg05263.html (887 words)

	Hand Coding » bttf
		I had preordered the Back to the Future DVDs, and they arrived today.
		However, Julie made me aware that there was a framing problem in the original release (a second run, supposedly in February, would correct it).
		You are currently browsing the archives for the bttf category.
	www.handcoding.com /archives/category/movies/bttf (315 words)

	Retrocactus.com » Nintendo (Site not responding. Last check: 2007-10-26)
		Considering the frenzy of preorders and stuff, I was still able to walk up to Walmart at 8am this morning and be the 51st in a lineup of people waiting to get inside and grab one of 54 units my local WM had.
		While waiting to get into EB to collect my preorder, there was a mini mob of people hoping to get one of the 10 extra units inside.
		It appears this will not be necessary now as they did indeed have extra units and was able to secure a preorder Wii as well as Zelda which will be in short supply (at least at this EB) as they got 28 copies in and most were spoken for already.
	www.retrocactus.com /blog/index.php/category/nintendo (1042 words)

	Citations: Duke Mathematical Journal - Birkhoff (ResearchIndex)
		The sum (coproduct) functor on Set is re ected into Pos to de ne the extension of X;Y from objects to morphisms:....
		The relevance to concurrency is that cardinals, as unordered sets, can be interpreted as concurrent schedules while ordinals and more generally chains constitute sequential schedules.
		Birkhoff s application was to the arithmetic of cardinals and ordinals, which he proposed to unify by regarding both as posets, with cardinals as discrete posets and ordinals as....
	citeseer.ist.psu.edu /context/503633/0 (1540 words)

	constant_function (Site not responding. Last check: 2007-10-26)
		Notice that every empty function, that is, any function whose domain equals the empty set, is included in the above definition vacuously, since there are no x and y in A for which f(x) and f(y) are different.
		For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely, if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant.
		In a connected set, a function is locally constant if and only if it is constant.
	maps.profreehosting.info /wiki/?title=Constant_function (364 words)

	Charm City Cards FAQ PAGE (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		For example, you may preorder 3 different items due to be released on 3 separate weeks, or a combination of current and upcoming releases--either way the UPS rate would show $10.
		The shipment would be made when the last preordered product on your order is received and in stock.
		Although impossible to stock every product, we usually are able to locate other sportscard boxes and sets that are not listed on our website within several days of a request.
	www.charmcitycards.com.cob-web.org:8888 /faq.htm (1738 words)

	Phish :: News
		The footage, set to a soundtrack of Vida Blue music, includes three consecutive years of the demo derby at the Champlain Valley Fair in Essex Junction, VT as well as the 2004 Enduro Race at Thunder Road, a two-hundred lap race around a quarter-mile track in which almost anything goes.
		The intermission between the first and second set will be between 60 and 80 minutes, and the intermission between the second and third sets will run anywhere from 40-50 minutes.
		Set one features a rare Stash>Kung>Stash and an extremely uncommon performance of one-time live standard Satin Doll, before which Trey pauses to thank the hard-working crew for their help on the tour.
	www.phish.com /news/index.php?year=2004 (11564 words)

	HAF: Detailed Table of Contents
		Sets, 3 -- Mathematical Language and Informal Logic, 3 -- Basic Notations for Sets, 11 -- Ways to Combine Sets, 15 -- Functions and Products of Sets, 19 -- ZF Set Theory, 25 2.
		Filters, Topologies, and Other Sets of Sets, 100 -- Filters and Ideals, 100 -- Topologies, 106 -- Algebras and Sigma-Algebras, 115 -- Uniformities, 118 -- Images and Preimages of Sets of Sets, 122 -- Transitive Sets and Ordinals, 122 -- The Class of Ordinals, 127 6.
		Concrete Categories, 208 -- Definitions and Axioms, 210 -- Examples of Categories, 212 -- Initial Structures and Other Categorical Constructions, 217 -- Varieties with Ideals, 221 -- Functors, 227 -- The Reduced Power Functor, 229 -- Exponential (Dual) Functors, 238 10.
	www.math.vanderbilt.edu /~schectex/ccc/excerpts/contents.html (994 words)

	iSkin eVo2 \| iLounge iPod Accessory Reviews
		Both eVo case sets include a considerable number of design innovations from Canada-based iSkin: their silicone rubber case moldings cover everything but an iPod’s controls, top ports, and screen, using different thicknesses of rubber to facilitate easy use of both under-the-eVo screen protectors and top-mounting iPod accessories.
		Taking these components together, the eVo 2 is unquestionably the most complete rubber case package currently sold for the 4G iPod.
		Contrasted with Power Support’s previously reviewed 4G Silicone Jacket Set ($28.00), the eVo 2 is slightly more expensive and doesn’t protect the iPod’s Click Wheel, but does include the detachable belt clip, more rugged screen protection, Dock Connector protection, and its own carrying case.
	www.ilounge.com /index.php/ipod/review/iskin-evo2-ipod (3638 words)

	Hand Coding » dvd
		Mr Bean: The Whole Bean is now available on DVD, as a 3-disc set (woo!).
		Much of the humor derives from the sheer cluelessness of the principals and the time-honored comedic tradition of rendering the dominant member of the team, in this case Rimmer, a bossy, self-absorbed twit who thinks he's a lot more impressive than he actually is. […]
		You are currently browsing the archives for the dvd category.
	www.handcoding.com /archives/category/dvd (1359 words)

	DS Fanboy
		Japan set to implode: Dragon's Quest IX is a DS exclusive
		Only a bare few new releases are set to brighten up shelves this week, and franchise licenses abound.
		The DS motion card has not yet been released, but is available for preorder at the very reasonable price of $25.
	www.dsfanboy.com /category/news (2223 words)

	Equivariant Homotopy Type of Categories and Preordered Sets (ResearchIndex) (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		Equivariant Homotopy Type of Categories and Preordered Sets (1999)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		6 Homotopy colimits in the category of small categories (context) - Thomason - 1979
	citeseer.ist.psu.edu.cob-web.org:8888 /villarroel-flores99equivariant.html (535 words)

	Amazon.com: The Theory of Quantaloids: Books: K I Rosenthal (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-26)
		Discover new releases in your favorite categories, popular pre-orders and bestsellers, exclusive author interviews and podcasts, special sales, and more.
		The basic theory, examples and construction are given and particular emphasis is placed on the free quantaloid construction, as well as on the perspective provided by enriched categories.
		In this chapter, after developing some necessary background material on quantales, enriched categories and sup-lattices, we present the definition of quantaloid, as well some of the basic properties.
	www.amazon.com.cob-web.org:8888 /exec/obidos/ASIN/0582294401 (818 words)

	[No title]
		I just started using Mail the way I use Gmail: anything that matched a filter got filed in the appropriate folder, and I could sort through other stuff left in my Inbox.
		Some Smart Folders take care of stuff that goes into more than one category.
		Of course, the point's fairly moot since I'm letting my.Mac subscription expire and that was the only thing I used Mail with anyway.
	arstechnica.com /journals/apple.ars/2006/10/10/5563 (666 words)

	TangognaT » comics
		Mouse Guard opens with the mysterious disappearance of a grain merchant who was traveling alone.
		Three intrepid members of the Mouse Guard named Lieam, Kenzie, and Saxon set out to investigate.
		David Petersen’s art is richly detailed, you can take a look at previews for the first and second issues on the Archaia Studios web site.
	www.tangognat.com /category/comics/page/4 (1247 words)

	HTBlog » Movies
		Once finishing the first half of the season on DVD (what the?), the second half is finally being released.
		The new two-disc sets of Star Wars: Episode IV—A New Hope, Star Wars: Episode V—The Empire Strikes Back and Star Wars: Episode VI—Return of the Jedi feature both the original and remastered film versions.
		Each street Sept. 12 (prebook Aug. 16) from 20th Century Fox Home Entertainment priced at $29.98 apiece and will be available only until Dec. 31.
	www.htblog.net /category/movies (1532 words)

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