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Topic: Product category theory

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	Product (category theory) - Wikipedia, the free encyclopedia
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces.
		Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
		The product construction given above is actually a special case of a limit in category theory.
	en.wikipedia.org /wiki/Product_(category_theory) (399 words)

	PlanetMath: category theory
		Category theory gives us tools for analyzing such functors: we can talk about natural transformations of functors, and in fact we can use these to assemble the category of functors from one category to another into a category, provided certain set-theoretic constraints are met (universes are a tool used to address these set-theoretic difficulties).
		The fundamental theorem of Galois theory is that the functor from a subgroup of the Galois group of a field to its fixed field is an equivalence of categories.
		This is version 4 of category theory, born on 2004-02-25, modified 2004-03-16.
	planetmath.org /encyclopedia/CategoryTheory.html (1655 words)

	Category theory - FreeEncyclopedia (Site not responding. Last check: 2007-11-06)
		Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories.
		Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
	openproxy.ath.cx /ca/Category_theory.html (2075 words)

	Category Theory
		Category theory is a general mathematical theory of structures and sytems of structures.
		Category theory reveals that many of these constructions are in fact special cases of objects in a category with what is called a "universal property".
		Indeed, from a categorical point of view, a set-theoretical cartesian product, a direct product of groups, a direct product of abelian groups, a product of topological spaces and a conjunction of propositions in a deductive system are all instances of a categorical concept: the categorical product.
	www.science.uva.nl /~seop/archives/win2003/entries/category-theory (3074 words)

	Product (category theory) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		Let C be a category and let be an (Click link for more info and facts about indexed family) indexed family of objects in C.
		The product construction given above is actually a special case of a (The greatest possible degree of something) limit in category theory.
		The product can be defined as the limit of any (Click link for more info and facts about discrete subcategory) discrete subcategory in C.
	www.absoluteastronomy.com /encyclopedia/p/pr/product_(category_theory).htm (611 words)

	Product (Site not responding. Last check: 2007-11-06)
		In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
		When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative.
		The dot product and cross product are forms of multiplication of vectors.
	www.city-search.org /pr/product.html (352 words)

	Category Theory (Reading Course)
		The product in Ens* is obtained by taking the cartesian product with the basepoint being the ordered pair; that is, if (A,a_0) and (B,b_0) are two sets with basepoint, then (A x B, (a_0,b_0)) is the product, in the sense of satisfying the diagram I gave in class.
		Category theory is a kind of network algebra which provides a general framework for describing mathematical objects and their interrelations.
		The standard example of a category is the family Ens of sets and functions with composition defined as ordinary composition of functions and the identity arrow is the identity function.
	www.georgetown.edu /faculty/kainen/category.html (1030 words)

	Discrete category - Wikipedia, the free encyclopedia
		In category theory, a discrete category is a category whose only morphisms are the identity morphisms.
		Any subcategory of a discrete category is discrete.
		The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.
	en.wikipedia.org /wiki/Discrete_category (97 words)

	18: Category theory, homological algebra
		Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry.
		While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology.
		A full, wide-ranging text on category theory is by Borceux, Francis: "Handbook of categorical algebra", 3 vol (1: Basic category theory; 2: Categories and structures; 3: Categories of sheaves) (Encyclopedia of Mathematics and its Applications, 50-2.) Cambridge University Press, Cambridge, 1994.
	www.math.niu.edu /~rusin/known-math/index/18-XX.html (286 words)

	Category Theory
		Category theory now occupies a central position not only in contemporary mathematics, but also in theoretical computer science and even in mathematical physics.
		However, it is still evolving and the precise meaning of category theory, that is what it is in the end about, remains to be fully clarified.
		After their 1945 paper, it was not clear that the concepts of category theory would be more than a convenient language and so it remained for approximately fifteen years.
	www.science.uva.nl /~seop/archives/win2004/entries/category-theory (7032 words)

	Categories and functors for the structural-phenomenol,ogical modeling
		Because in the definition of a category, it is not required that its objects should be sets with elements [11], that is usual mathematical objects, a category with its objects being phenomenological senses is called phenomenological category.
		Although such categories may be considered at a very abstract level, the practice of categories and functors, used mostly in the mathematical domain, has shown, as observed before, that the best and fruitful results may be obtained for particular domains of mathematics (for Abelian groups, topological spaces etc.).
		The feasibility condition is essential for the adaptation of the theory of categories to a mathematical structural-phenomenological theory of categories.
	www.racai.ro /~dragam/Categories.html (2762 words)

	MATHS: Category Theory
		Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic.
		A Category is a mixture of an algebra and a directed graph.
		Category theory shows that in most known algebras and logistic systems, there is a way to construct an equivalent.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	Category Theory for Computing Science
		Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
		Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another.
		In this chapter, a method is given for translating between finite product sketches and the formalism of signatures and equations used in traditional universal algebra.
	www.cwru.edu /artsci/math/wells/pub/ctcs.html (1730 words)

	Product (mathematics) (Site not responding. Last check: 2007-11-06)
		In mathematics, a product is the result of multiplying, or an expression that identifies factorss to be multiplied.
		The dot product and cross product are forms of multiplication of vectorss.
		Products in rings and fieldss of many kinds.
	www.sciencedaily.com /encyclopedia/product__mathematics_ (193 words)

	Netscape Search Category - Theory
		Java Music Theory Suite of Java applets designed to help students of music theory improve their proficiency at basic music theory skills.
		Texas Institute of Theory Devoted entirely to the promulgation of music theory in the Western European tradition.
		Theory of Music with Ted Kirk Resources for learning and teaching music theory to grade 5 of the Associated Board (UK).
	open.netscape.com /Arts/Music/Theory (549 words)

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		An antioxidant enzyme that protects the energy production of the cells prolong the life of mice by 20%.
		No danger in taking the vitamins C and -E. A number of the world's leading scientists establish that the vitamins C and -E are harmless even in large doses.
		At the same time, the theory that the two vitamins prevent chronic illness is still very much alive.
	www.vitaviva.com /en/Shop/Product_List.1.12.aspx (303 words)

	Category Theory
		By the early seventies, the concept of adjoint functors was considered to be the central concept of category theory.
		For it is in his thesis that Lawvere proposed the idea of developing the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of theories, that is the logical aspects of mathematics.
		Even though the concept of a topos was presented in the sixties in the context of algebraic geometry, it was certainly Lawvere and Tierney's work on the elementary axiomatization of the concept, published in the early 1970s, which gave to the notion its foundational status and impetus.
	plato.stanford.edu /entries/category-theory (7029 words)

	Product and Coproduct
		The product in a category is inspired by the direct product of groups, rings, and modules.
		If the category is concrete, and the index is finite, the product is simply the cartesian product, with the usual component projections.
		In the category of abelian groups, rings, or r modules, the coproduct of finitely many components is the same as the product.
	www.mathreference.com /cat,prod.html (1156 words)

	Newsgroops - Re: Category theory question (Site not responding. Last check: 2007-11-06)
		and the sum is a direct summand of the product.
		This is the case in all abelian categories admitting infinite sums and
		being a sujection from the product to the sum arises.
	www.newsgroops.org /group/sci.math/article-281991.html (309 words)

	Product (category theory): Definition and Links by Encyclopedian.com - All about Product (category theory) (Site not responding. Last check: 2007-11-06)
		Product (category theory): Definition and Links by Encyclopedian.com - All about Product (category theory)
		Suppose C is a category, I is a set, and for each i in I, an object X
		) needs to have a product, but if it does, then the product is unique in a strong sense: if p
	www.encyclopedian.com /pr/Product-(category-theory).html (343 words)

	Category theory preprints 2002
		The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.
		We extend Shrimpton's investigations on the morphism-digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal.
		In this paper we present applications of freerange mapping spaces to the theories of cofibrations, the cohomology of fibrations, sectioned fibrations, identifications and Moore-Postnikov factorizations.
	www.informatics.bangor.ac.uk /public/mathematics/research/preprints/02/cathom02.html (838 words)

	Theory and Applications of Categories
		Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads
		The alternation hierarchy for the theory of $\mu$-lattices
		On the monadicity of categories with chosen colimits
	www.tac.mta.ca /tac (989 words)

	[Inquiry] Re: Category Theory (Site not responding. Last check: 2007-11-06)
		A 'category' is a graph with two additional functions:
		In treating a category C, we usually
		category is a monoid for the product
	stderr.org /pipermail/inquiry/2003-July/000623.html (298 words)

	category theory - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "category theory" is defined.
		Category Theory : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include category theory: baire category theory, category theory object, list of category theory topics, product category theory
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=category+theory (108 words)

	Product Category : Multichannel Analyzers (MCAs) (Site not responding. Last check: 2007-11-06)
		Canberra’s Nuclear Products Group offers a complete line of personal computer-based, and DEC Alpha computer based multichannel analyzers to meet the ever changing needs of the nuclear instrument marketplace.
		The current Canberra MCA product line consists of multichannel analyzers capable of simultaneous data acquisition, display, analysis and input/output from one or more ADCs.
		The use of microprocessors in their architecture allows extensive and sophisticated data analysis to be performed within the MCA, so that final results may be displayed and printed.
	ww2.canberra.com /pcatalog.nsf/ProductCatalog/BC288134136C7AD2852568620068A9A2?OpenDocument&area=product (1792 words)

	Clearing up the market cycle... best Product Category Theory (Site not responding. Last check: 2007-11-06)
		If Periodia value is 26, that means that from the vary last maximum of its passed 26 ticks of time and we have "period length"/2-26 ticks of time to reverse point.
		Although asthma may be commonly regarded as a condition affecting people during the daytime, a new survey shows nearly half (45 percent) of persistent asthma sufferers report their worst asthma symptoms occur in the evening and nighttime.
		smash" product can be ignored - it was about a topological problem and is not needed (or correct) for category theory...
	ascot.pl /th/Fourier5/Product-Category-Theory.htm (765 words)

	week202
		A rig category is basically the most general sort of category in which we can "add" and "multiply" as we do in a ring - but without negatives, hence the missing letter "n".
		He gracefully leads the reader from the very basics of category theory straight to the current battle front of weak n-categories, emphasizing throughout how operads automatically take care of the otherwise mind-numbing thicket of "coherence laws" that inevitably infest the subject.
		It relates the category whose objects are 2-manifolds with a circle as boundary, and whose morphisms are 3-manifolds with corners going between these, to a braided monoidal category "freely generated by a quasitriangular Hopf algebra object".
	math.ucr.edu /home/baez/week202.html (4106 words)

	OUP: Category Theory 1991: Seely (Site not responding. Last check: 2007-11-06)
		As category theory approaches its first half-century, it continues to grow, finding new applications in areas that would have seemed inconceivable a generation ago, as well as in more traditional areas.
		The language, ideas, and techniques of category theory are well suited to discovering unifying structures in apparently different contexts.
		Occasionally, due to the nature of some contractual restrictions, we are unable to ship a specific product to a particular territory.
	www.oup.co.uk /isbn/0-8218-6018-6 (296 words)

	Category Theory for Computer Science (Site not responding. Last check: 2007-11-06)
		Cartesian closed categories and the simplytyped lambda calculus.
		universal constructions in Cat (the category 1 with one object and one arrow, the empty category 0, product category, sum of categories)
		Using Category Theory to Design Implicit Conversions and Generic Operators.
	www.daimi.au.dk /~nygaard/CTfCS (620 words)

	product - OneLook Dictionary Search
		Product : eyefortransport e-commerce transportation glossary [home, info]
		Phrases that include product: gross national product, gross domestic product, by product, inner product, universal product code, more...
		Words similar to product: intersection, merchandise, production, wares, mathematical product, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=product (414 words)

	Category Theory
		Prove a theorem about groups, and it can be applied in many different situations, like calling up a program with different parameters.
		Category theory is the greatest generalization of them all.
		It is so general, it is sometimes called generalized abstract nonsense.
	www.mathreference.com /cat,intro.html (689 words)

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