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Topic: Category of sets

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	Category of sets - Biocrawler
		In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions.
		The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B.
		Set is thus a topos (and in particular cartesian closed).
	www.biocrawler.com /encyclopedia/Category_of_sets (264 words)

	Category of sets (Site not responding. Last check: 2007-10-19)
		In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions.
		The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.
		The powerobject of a set A is given by its power set, and the exponential object of thesets A and B is given by the set of all functions from A to B.
	www.therfcc.org /category-of-sets-178038.html (240 words)

	Category of sets - Wikipedia, the free encyclopedia
		In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions.
		Because of Russell's paradox, which shows assuming the existence of the set of all sets leads to a contradiction, the object class of Set is a proper class, and thus the category is large.
		The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B.
	www.wikipedia.org /wiki/Category_of_sets (309 words)

	PlanetMath: category of sets
		The category of sets has as its objects all sets and as its morphisms functions between sets.
		Alternately one can specify a universe, containing all sets of interest in the situation, and take the category to contain only sets in that universe and functions between those sets.
		This is version 1 of category of sets, born on 2002-02-10.
	www.planetmath.org /encyclopedia/CategoryOfSets.html (104 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.
		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.
		Category theory reveals that many of these constructions are in fact certain objects in a category having a "universal property".
	plato.stanford.edu /entries/category-theory (11810 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".
		This simply means that, given two categories C and D, a functor F from C to D, should send objects of C to objects of D and morphisms of C to morphisms of D in such a way that composition of morphisms in C is preserved, i.e.
	setis.library.usyd.edu.au /stanford/archives/fall1997/entries/category-theory (3066 words)

	Structuralism, Category Theory and Philosophy of Mathematics
		Category theory is the language best suited for this type of representation because it avoids the incommensurability problems which result from the Tarskian semantics essential to mathematical logic and model- theory for which satisfaction relations and truth definitions can only be defined for a specific language and the structure used to explicate the semantics.
		This is due to the fact that set theory is extensional, and the combinatorial aspects of mathematics, which is concerned with the finitely presented properties of the inscriptions of the formal language is intensional.
		Category theory furnishes such a formulation through the concept of topos, and its formal counterpart, local set theory.[Bell,238] Any topos may be regarded as a mathematical domain of discourse or 'world' in which mathematical concepts can be interpreted and mathematical constructions performed.
	www.mmsysgrp.com /strctcat.htm (7237 words)

	Math Forum Discussions
		Indeed many categories of structures have lots of properties, but the category of sets enjoyes a lot of these properties.
		While a set is 'merely' just a set and other mathematical structures which are based on sets appear to be a set with 'much more', the category of 'just sets' is very reach.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/message.jspa?messageID=3927156&tstart=0 (289 words)

	Concrete category (Site not responding. Last check: 2007-10-19)
		In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition offunctions.
		Most categories considered in everyday life are concrete; examples are the category of topological spaces with continuous maps asmorphisms or the category of groups with group homomorphisms as morphisms.
		In the formalapproach, a concrete category is defined as a category together with a faithful functor into the category of sets.
	www.therfcc.org /concrete-category-206642.html (170 words)

	Avon 4 Sale Avon4sale_Avon Men's and Women's Sets Page. (Site not responding. Last check: 2007-10-19)
		The set was made in 1986 and it is in Mint Condition.
		It is a father and son set and made of ceramic in 1987.
		This set consists of two soaps with roosters embossed in the soap and the thermometer tin top box, that has some tiny rust spot on the top.
	avon4sale.zoovy.com /category/avon_sets (827 words)

	MATHS: Category Theory (Site not responding. Last check: 2007-10-19)
		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.
		By analogy with sets and mappings, one end of the arrow we call the Domain and the other the Codomain, but we notice that the is no necessity for the arrows to actually be mappings.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	Image (category theory) - Wikipedia, the free encyclopedia
		In the category of sets the image of a morphism
		In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.
		In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism f can be expressed as follows:
	en.wikipedia.org /wiki/Image_(category_theory) (158 words)

	Category Theory and Homotopy Theory (Site not responding. Last check: 2007-10-19)
		Category theory was introduced in 1947 to give a richer language than that of set theory, which would be better able to express the structures of homotopy and homology theory then being revealed in the work of Cartan, Eilenberg, Mac Lane, Whitehead and others.
		In addition to the objects in a category (corresponding to the elements in a set), one also has arrows or "morphisms" between them.
		The basic areas of research in category theory at Bangor are directed towards achieving a greater understanding of the categorical structure and interrelationships between the various objects studied by algebraic topology and homological algebra.
	www.informatics.bangor.ac.uk /public/mathematics/research/cathom/cathom1.html (222 words)

	Citations: An elementary theory of the category of sets - Lawvere (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		An elementary theory of the category of sets.
		F.W. Lawvere (1964) An elementary theory of the category of sets, Proc.
		F.W.Lawvere, An elementary theory of the category of sets, Proc.
	citeseer.ist.psu.edu /context/96391/0 (1451 words)

	[No title]
		Category theorists are keen on statements to the effect that structures are defined by their universal properties.
		The nonsensical things in set theory seem to arise from a basic nonsensical postulate: that there is a single fundamental relation "element of" on the mathematical universe that is sufficient to determine the nature of every mathematical object.
		It's the null set, and the singleton of the null set, and [and on.
	www.mta.ca /~cat-dist/catlist/1999/set-memb-func-comp (15844 words)

	Cat::Set -- the category of sets of complex numbers (Site not responding. Last check: 2007-10-19)
		Sets of this category allow set-theoretic operations as well as pointwise arithmetical operations.
		By default, any operation of sets is defined, but returns unevaluated since the arithmetical or set-theoretic expression cannot be simplified.
		By overloading this method in a particular domain, the behaviour of sets changes whenever a special function is applied to them.
	www.sciface.com /STATIC/DOC25/eng/Cat/Set.shtml (793 words)

	PlanetMath: representable functor (Site not responding. Last check: 2007-10-19)
		and the category of sets is representable if there is an object
		the category of schemes, to address one of Grothendieck's primary interests) and embed it into a category of functors (as above).
		Cross-references: decide, natural transformations, theorem, primary, schemes, property, ring, adjoint, Yoneda Lemma, isomorphic, category of sets, category, between, contravariant functor
	planetmath.org /encyclopedia/RepresentableFunctor.html (217 words)

	CMS Summer 2002 Meeting
		This is at once a generalization of the category of sets and relations, and the category of coherence spaces.
		Categories of games are usually employed as the basis for providing a semantics for a free category carrying a specified structure.
		In Ner(B), each of the sets of simplices forms the set of objects of a natural category structure for which the face and degeneracy maps become functorial and the 0-simplices become the objects of a discrete category.
	www.cms.math.ca /Events/summer02/abs/ct.html (3009 words)

	www.goldnsilver.com: Category: 'Proof Sets'
		Beginning in 1999, sets include all five of the state quarters for the year.
		This years proof set includes the two new Westward Journey Nickels, as well as the dime, Kennedy half, penny, Sacagawea Dollar, and five state quarters.
		This years 11 piece set includes the final two nickels in the Westward Journey series, as well as the 5 State Quarters for the year.
	www.goldnsilver.com /blanksite/category.asp?dept_id=6 (239 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.
		These sketches provide a formalism for universal algebra that provide a natural definition of models in categories other than the category of sets, and being based on graphs, they also incorporated multisortedness into the definition in an intrinsic way.
	www.cwru.edu /artsci/math/wells/pub/ctcs.html (1730 words)

	Cocktail Sets - Picnic Fun
		Choose from our wide range of sets that include everything you need for a party on the go.
		These unique, practical sets are a great choice for parties, gatherings, or holidays.
		The convenient cases can be easily stored away at home or carried with you wherever you choose to take it.
	www.picnicfun.com /category/cocktail-sets (119 words)

	[No title]
		Let D be the category whose objects are set-valued bifunctors on A (contravariant on the first variable, covariant on the second) and whose maps are the dinatural transformations.
		From: Peter Freyd [Let D be the category whose objects are] set-valued bifunctors on A (contravariant on the first variable, covariant on the second) Two definitional/notational suggestions: 1.
		If A is a groupoid then the category composed of dinaturals between bifunctors from A to B is equivalent to the category composed of natural transformations between covariant functors from A to B.
	www.mta.ca /~cat-dist/catlist/1999/dinatural (731 words)

	Lusternik-Schnirelmann Category and Simplicial Sets (Site not responding. Last check: 2007-10-19)
		Since all of the notions which are required to define the Lusternik-Schnirelmann category of a topological space exist for simplicial objects, the various definitions of Lusternik-Schnirelmann category can be used to provide similar definitions for simplicial objects.
		Furthermore, there exist well known adjoint functors between topological spaces and simplicial sets, and it is reasonable to ask how these new definitions compare with the topological definitions.
		The goal of this work is to prove that the various definitions of Lusternik-Schnirelmann category all yield the same result for simplicial objects and that this value is compatible with the definitions for topological spaces.
	pc75666.math.cwu.edu /%7Emontgomery/scholar/1996/0105-t-joint_abs.html (100 words)

	Citations: Using category theory to design implicit conversions and generic operators - Reynolds (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		These coercions are also present in the speci cations that other authors have used in their work on behavioral subtyping [2, 5, 10, 26] In our work, the coercions take the from of equations, and we show that if all the coercion equations are behavioral consequences of the given equational....
		The intuition behind this is that a nite set W is the set of labels for memory variables that are currently allocated, called a world.
		uses category theory to investigate the problems of the interaction of coercions (implicit conversions) and generic operators (also called overloaded operators) He concludes with the global requirement that all possible coercions and generic operators have to commute to get a sound system.
	citeseer.ist.psu.edu /context/35117/0 (3724 words)

	Abstract: A New Category for Semantics (Site not responding. Last check: 2007-10-19)
		Generally, the effort has been to create categories of domains that are cartesian closed (that is, have products and function spaces interpreting typed lambda calculus) and permit solutions to domain equations (that is, interpret recursive domain definitions and perhaps untyped lambda calculus).
		This category is not only cartesian closed, but it is locally cartesian closed (that is, it has dependent sums and products).
		The logic for this category is intuitionistic and can be explained by a form of the realizability interpretation.
	www.cs.cmu.edu /Groups/LTC/papers/EQULogic_abstract.html (326 words)

	www.goldnsilver.com: Category: 'Medallion Sets'
		This set includes one of our Anniversary silver medallions and a Silver American Eagle in a hot stamped box.
		This set includes one of our Graduation Silver Medallions and a Silver American Eagle.
		This set includes one of our Mothers Day Silver Medallions and a Silver American Eagle.
	www.goldnsilver.com /category.asp?dept_id=127 (238 words)

	Tool Sets (Site not responding. Last check: 2007-10-19)
		Superbly versatile socket set with a variety of 1/4" and 1/2" sockets and accessories.
		This set is suitable for 35 different fastener sizes with a total of 24 sockets....
		STANLEY SOCKET SET - 1/4,3/8,1/2IN SQUARE DRIVE 75 Piece Combination Socket and Bit Set comprising of a range of sockets and accessories covering popular sizes in 1/4in, 3/8in and 1/2in...
	www.discount-warehouse.org.uk /tools/Tool-Sets.shtml (429 words)

	Smyrna Coin Shop - www.smyrnacoin.com 1999, 2000, 2001, 2002 and 2003 State Quarter Sets...B.U. and Proof
		Complete your State Quarter collection with these beautiful Proof strike coins and be among the first to see all the 2003 State Quarter designs!!...
		Complete set of 2001 Proof State Quarters from the San Francisco Mint ("S" Mint mark)...
		This set includes all five 2004 Silver Proof State Quar...
	www.smyrnacoin.com /category/state_quarter_sets (503 words)

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