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	Cartesian closed category - Wikipedia, the free encyclopedia
		The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category.
		In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed.
		Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory.
	en.wikipedia.org /wiki/Cartesian_closed (910 words)

	Cartesian closed category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		In (additional info and facts about category theory) category theory, a category is cartesian closed if, roughly speaking, any (additional info and facts about morphism) morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors.
		The category of all (additional info and facts about directed graphs) directed graphs is cartesian closed; this is a functor category as explained under (additional info and facts about functor category) functor category.
		Certain cartesian closed categories, the (additional info and facts about topoi) topoi, have been proposed as a general setting for mathematics, instead of traditional (The branch of pure mathematics that deals with the nature and relations of sets) set theory.
	www.absoluteastronomy.com /encyclopedia/c/ca/cartesian_closed_category.htm (1096 words)

	Cartesian closed category: Definition and Links by Encyclopedian.com - All about Cartesian closed category
		The term "cartesian closed" is used because one thinks Y×X as akin to the cartesian product of two sets.
		In cartesian closed categories, a "function of two variables" can always be represented as a "function of one variable".
		In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continous maps nor the category of smooth manifolds with smooth maps is Cartesian closed.
	www.encyclopedian.com /ca/Cartesian-closed-category.html (414 words)

	cartesian closed category (Site not responding. Last check: 2007-10-21)
		The category Set of all sets, with functionss as morphisms, is cartesian closed.
		In other contexts, this is known as currying; it has led to the realization that lambda calculus can be formulated in any cartesian closed category.
		Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics.
	www.yourencyclopedia.net /cartesian_closed_category.html (832 words)

	Cartesian closed category (Site not responding. Last check: 2007-10-21)
		The category of all directed graphs is cartesian closed; this is afunctor category as explained under functor category.
		In algebraic topology, cartesian closed categories areparticularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor thecategory of smooth manifolds with smooth maps is cartesian closed.
		Certain cartesian closed categories, the topoi, have been proposed as a generalsetting for mathematics.
	www.therfcc.org /cartesian-closed-category-178037.html (746 words)

	Cartesian closed category (Site not responding. Last check: 2007-10-21)
		The category of all directed graphs is cartesian closed; this is a category as explained under functor category.
		In algebraic topology cartesian closed categories are particularly easy work with and it is regrettable that the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed.
		Certain cartesian closed categories the topoi have been proposed as a general for mathematics.
	www.freeglossary.com /Exponential_(category_theory) (1180 words)

	Stable Domain Theory (Site not responding. Last check: 2007-10-21)
		Separately, Berry [78] constructed a cartesian closed category whose morphisms preserve directed joins and connected meets, whilst Diers [79] considered similar functors independently in a study of categories of models of disjunctive theories.
		Berry's proof of cartesian closure (using the trace factorisation, which also occurs in Diers' work and is discussed in [Taylor 88]) and more direct proofs by Coquand [88] and Lamarche [88] use two additional hypotheses, strong finiteness and distributivity (of finite meets over finite joins).
		We use the results of a previous paper, where it was shown that functors with adjoints on each slice admit a factorisation, known to Berry as the trace and to Diers as a spectrum.
	www.cs.man.ac.uk /~pt/stable/index.html (1137 words)

	Cartesian closed topological hull of the construct of closure spaces (Site not responding. Last check: 2007-10-21)
		Cartesian closed topological hull of the construct of closure spaces
		A cartesian closed topological hull of the construct CLS of closure spaces and continuous maps is constructed.
		Secondly, within this extension L the cartesian closed topological hull L* of CLS is characterized as a full subconstruct.
	www.emis.ams.org /journals/TAC/volumes/8/n18/abstract.html (130 words)

	Topology in Computer Science
		Moreover, he proved that every cartesian closed category of algebraic domains is contained in one of these.
		Moreover, given a cartesian closed category of algebraic domains, the category comprising the retract of its objects is cartesian closed again.
		As shown by Achim Jung in [Jun90] the picture is similar to the algebraic case: There are exactly two maximal cartesian closed categories of continuous domains, that of FS-domains and that of continuous L-domains, and every cartesian closed category of continuous domains is contained in one of these.
	www.informatik.uni-siegen.de /theo/TopCS.html (1549 words)

	Cartesian Closed Topological Hull Of The Construct Of Closure Spaces (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: A cartesian closed topological hull of the construct Cls of closure spaces and continuous maps is constructed.
		First a cartesian closed extension L of Cls is obtained.
		Secondly, within this extension L the cartesian closed topological hull L # of Cls is characterized as a full...
	citeseer.ist.psu.edu /claes01cartesian.html (406 words)

	Citations: A note on inconsistencies caused by fixpoints in a cartesian closed category - Huwig, Poign'e (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		, a cartesian closed category with fixpoints and finite sums is equivalent to the category with one object and one arrow.
		that a cartesian closed category with finite sums and a fixpoint operator is inconsistent, that is, it is equivalent to the category consisting of one object and one arrow.
		a cartesian closed category with finite sums together with a fixpoint operator is shown to be inconsistent, that is, it is equivalent to the category consisting of one object and one map.
	citeseer.ist.psu.edu /context/142168/0 (2381 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Date: Sat, 14 Dec 1996 16:33:22 -0400 (AST) Subject: Cartesian Closed arrow categories Date: Fri, 13 Dec 96 16:01:11 EST From: Kathryn_Van_Stone@POP.CS.CMU.EDU Does anyone what conditions are necessary in a category C for its arrow category to be Cartesian Closed.
		Let's agree that a category A is cartesian closed when it has finite products and each functor a x - : A --> A has a right adjoint [a,-].
		Assuming that by "arrow category" you mean the functor category [2,C], this is a special case of the question of when a category obtained by Artin glueing is cartesian closed.
	www.mta.ca /~cat-dist/catlist/1999/ccc-arrow (228 words)

	Practical Foundations of Mathematics
		Cartesian closed categories of domains The category of sets and total functions is the fundamental interpretation of the typed
		The first three parts were proved in Propositions 3.5.1 and 3.5.5, but it is the notion of cartesian closed category which makes sense of the collection of facts in Section 3.5.
		At the end of the next section we shall show that categories themselves may be considered as domains and form a cartesian closed category.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s47.html (1743 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		As far as I am aware these constructions were originally given by Lambek [lam74], proofing the corresponding deduction theorem for cartesian closed categories, and revised by Lambek and Scott [lam&sco86].
		Although the above result is well known it seems that the result of adjoining an intedeterminate object to a cartesian closed category, in particular it's proof, is not so well known.
		D, where D is a cartesian (closed) category, F: C -> D is a cartesian closed functor, and D is in Obj(D).
	www.mta.ca /~cat-dist/catlist/1999/adj-indet (439 words)

	open and closed sets (Site not responding. Last check: 2007-10-21)
		What is the difference between open and closed captioning?...
		The Grid-Cube Two-Piece Property Closed Sets in the Plane With the GTPP...
		Automatic detection of open and closed separation and attachment lines...
	www.scienceoxygen.com /math/359.html (242 words)

	FLoC '02 - DOMAIN Sunday July 21st
		Dana Scott showed in 1970 that there are many cartesian closed categories containing arbitrarily large objects which are isomorphic to their own self-exponential.
		The later is also equivalent to that the set C(P) of all Scott closed subsets sets of P is a completely distributive lattice.
		In [1], Amadio and Curien raised the question of whether the category of stable bifinite domains is the largest cartesian closed full sub-category of the category of ω-algebraic meet-cpos with stable functions.
	floc02.diku.dk /DOMAIN/Sunday.html (1256 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The intuition behind cartesian closedness is that any function from C \times A to B is supposed to correspond to a function from C to B^A. But this correspondence doesn't work in NF, because it isn't compatible with stratification.
		Similar considerations show that Set cannot be shown to be Cartesian closed using some nonstandard approach.
		If Set were Cartesian closed, there would be a one-to-one correspondence between arrows from V to V (functions from the universe into the universe) and arrows from 1 to V^V (functions with domain a fixed singleton and range the exponential object).
	math.boisestate.edu /~holmes/holmes/fomletter17.txt (207 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		Cartesian closure is usually defined only for categories possessing all finite limits (see, for example, MacLane [19]).
		Of course, whenever the cartesian product of a pair of objects exists in a full subcategory K of DCPO, it is easy to prove that this object is the K-product of the pair in K.
		The counterexample demonstrating that Algpos is not closed with respect to the formation of function spaces now deserves some careful attention as it sheds light on what restrictions we must place on algebraic posets to obtain the desired closure.
	www.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	Cartesian closed category (Site not responding. Last check: 2007-10-21)
		The category of all vector spaces over some fixed field isn't cartesian closed, neither is the category of all finite-dimensional vector spaces.
		The category of abelian groups isn't cartesian closed, for the same reason.
		The category of topological spaces isn't cartesian closed.
	www.explainthis.info /ca/cartesian-closed-category.html (570 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		19 October 2004 4:00 - 5:30 Nicola Gambino Wellfounded trees in locally cartesian closed categories Abstract: The correspondence between cartesian closed categories and simple type theories was extended in [1], where it was shown how locally cartesian closed categories relate to dependent type theories.
		The connections between type theories and categories were recently taken further with the introduction of a categorical counterpart of the type-theoretic notion of W-type [2].
		In the first part of my talk, I will present some joint work with Martin Hyland describing some of the consequences of the assumption of the existence of wellfounded trees in a locally cartesian closed category.
	www.math.mcgill.ca /rags/seminar/gambino.txt (231 words)

	Linear logic and -autonomous categories (Site not responding. Last check: 2007-10-21)*
		A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed lambda-calculus.
		The linear structure amounts to a -autonomous category: a closed* symmetric monoidal category G with finite products and a closed involution.
		, is a cotriple on G which carries the canonical comonoid structure on A with respect to cartesian product to a comonoid structure on !A with respect to tensor product.
	www.math.mcgill.ca /rags/nets/llsac.abstract.html (75 words)

	On the unification problem for Cartesian closed categories (Site not responding. Last check: 2007-10-21)
		Cartesian closed categories (CCC's) have played and continue to play an important role in the study of the semantics of programming languages.
		An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce and Longo leads to seven equalities.
		It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages.
	www.cl.cam.ac.uk /Research/HVG/ARG_Talks/abstracts/abstract_930520.html (163 words)

	Encyclopedia: Category theory (Site not responding. Last check: 2007-10-21)
		Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
		The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories.
	www.nationmaster.com /encyclopedia/category-theory (4301 words)

	MathAction and Axiom Tuples Products And Records
		But Cartesian Product is implemented as a Record and the domain Record is a primative in Axiom.
		In the case of Axiom I think that it is highly significant that "The category Cat of all small categories (with functors as morphisms) is cartesian closed;".
		But very briefly one answer to your question could be: CCC is the smallest and simplist kind of category in which we should expect to be able to do all the symbolic computation that is possible on a computer.
	www.axiom-developer.org /zope/mathaction/TuplesProductsAndRecords (6939 words)

	Lambek and Scott: Introduction to higher order categorical logic (Site not responding. Last check: 2007-10-21)
		Given any Galois correspondence, we may pick the subpreorder of A of closed a (i.e., GFa is isomorphic to a) and subpreorder of B of open b (i.e., FGb is isomorphic to b).
		A category with finite products (including a terminal object) is cartesian closed if for every B, -xB has a right adjoint -^B. Examples include Sets, presheaf categories (Sets^X for small categories X), any topos, and so on.
		The cartesian closed category generated by a typed lambda calculus.
	www.andrew.cmu.edu /~cebrown/notes/lambekscott.html (4587 words)

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