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Topic: Cartesian closed category

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Category theory

	Facts about topic: (Cartesian closed category) (Site not responding. Last check: 2007-10-22)
		These categories are particularly important in mathematical logic (Any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity) and the theory of programming (Creating a sequence of instructions to enable the computer to do something).
		The category Set of all set (A group of things of the same kind that belong together and are so used) s, with function (A mathematical relation such that each element of one set is associated with at least one element of another set) s as morphisms, is cartesian closed.
		The category of all vector space (additional info and facts about vector space) s over some fixed field (A piece of land cleared of trees and usually enclosed) is not cartesian closed, neither is the category of all finite-dimensional (additional info and facts about finite-dimensional) vector spaces.
	www.absoluteastronomy.com /encyclopedia/c/ca/cartesian_closed_category.htm (1094 words)

	Encyclopedia: Category theory
		Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
		Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, in connection with algebraic topology.
		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.
	www.nationmaster.com /encyclopedia/category-theory (476 words)

	Cartesian closed category: Definition and Links by Encyclopedian.com - All about Cartesian closed category
		If C is a small category, then the category of all functors from C to Set (with natural transformations as morphisms) is a cartesian closed category.
		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)

	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-10-22)
		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.
		Another important duality occurs in functional analysis: the category of commutative C-algebras with identity is contravariantly equivalent to the category* of compact Hausdorff spaces.
	www.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	Encyclopedia: Cartesian closed category
		In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object.
		The category Set of all (A group of things of the same kind that belong together and are so used) sets, with (A mathematical relation such that each element of one set is associated with at least one element of another set) functions as morphisms, is cartesian closed.
		The category of all (Click link for more info and facts about vector space) vector spaces over some fixed (A piece of land cleared of trees and usually enclosed) field is not cartesian closed, neither is the category of all (Click link for more info and facts about finite-dimensional) finite-dimensional vector spaces.
	www.nationmaster.com /encyclopedia/Cartesian_closed-category (472 words)

	Topos - Wikipedia, the free encyclopedia
		In the mathematical field of category theory, a topos is a type of category that behaves like the category of sheaves of sets on a topological space.
		Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space.
		For instance, the category of all directed graphs is a topos.
	en.wikipedia.org /wiki/Topos (1180 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.
		Set is thus a topos (and in particular cartesian closed).
	www.wikipedia.org /wiki/Set_(category_theory) (330 words)

	Category theory (Site not responding. Last check: 2007-10-22)
		Any preordered set (P, ≤) forms a small category, where the objects are the members of P, and the morphisms are arrows pointing from x to y when x ≤ y.
		These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies.
		It is a natural question to ask, under which conditions two categories can be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category.
	www.sciencedaily.com /encyclopedia/category_theory (3261 words)

	Category theory (Site not responding. Last check: 2007-10-22)
		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 algebraC(X) of all real-valued continuous functions on that space.
		A category is called cartesian closed if ithas finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just oneof the factors.
		A topos is a certain type of cartesian closed category in which all of mathematicscan be formulated (just like classically all of mathematics is formulated in the category of sets).
	www.therfcc.org /category-theory-21511.html (2892 words)

	Cartesian closed category (Site not responding. Last check: 2007-10-22)
		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.sciencedaily.com /encyclopedia/cartesian_closed_category (843 words)

	Encyclopedia topic: Product (category theory) (Site not responding. Last check: 2007-10-22)
		Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
		Let C be a category and let be an indexed family (additional info and facts about indexed family) of objects in C.
		Cartesian closed category (additional info and facts about Cartesian closed category)
	www.absoluteastronomy.com /encyclopedia/p/pr/product_(category_theory).htm (551 words)

	Cartesian closed category - Wikipedia, the free encyclopedia
		For the first two conditions above, it’s the same to require that any finite (possibly empty) family of objects of C admit a product in C, because of the natural associativity of the categorical product and noticing that the empty product in a category is nothing but the terminal object of that category.
		If X is a topological space, then the open sets in X form the objects of a category O(X) for witch there's a unique morphism from U to V if U is a subset of V and no morphism otherwise.
		The renowned computer scientist John Backus has advocated a variable-free notation, or Function-level programming, which in retrospect bears some similarity to the internal language of cartesian closed categories.
	www.wikipedia.org /wiki/Cartesian-closed_category (928 words)

	Category theory - Biocrawler (Site not responding. Last check: 2007-10-22)
		The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures.
		Such a process is called a functor, and it associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.
		Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945.
	www.biocrawler.com /biowiki/Category_theory (2428 words)

	Goldblatt. Topoi: The Categorial Analysis of Logic (Site not responding. Last check: 2007-10-22)
		These categories are examples of preorders: categories in which there is at most one arrow between any two objects (identity corresponds to reflexivity and composition corresponds to transitivity).
		A category is complete if it has all limits, co-complete if it has all colimits, bi-complete if it is complete and co-complete, finitely complete if it has all finite limits, finitely co-complete if it has all finite colimits, and finitely bi-complete if it is finitely complete and finitely co-complete.
		For Cartesian closed categories, we have 0xA is initial for all A, an arrow A->0 implies A is initial, we can only have a zero object if the category is degenerate (all objects are isomorphic), any arrow 0->A is monic, A^1 is isomorphic to A, A^0 and 1^A are terminal.
	www.andrew.cmu.edu /~cebrown/notes/goldblatt.html (7165 words)

	Cartesian closed category (Site not responding. Last check: 2007-10-22)
		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)

	Cartesian closed topological hull of the construct of closure spaces (Site not responding. Last check: 2007-10-22)
		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.univie.ac.at /EMIS/journals/TAC/volumes/8/n18/8-18abs.html (141 words)

	Centre of Australian Category Theory, Macquarie University :: Projects
		Summary: Category theory is a branch of mathematics concerned with transformation and composition.
		Higher-dimensional categories are complex structures that are currently gaining a lot of attention from mathematicians, physicists and computer scientists because of developing applications in those fields.
		He furthermore constructed an action of a categorical analogue of the little n-cubes operad on the category of extensions in a monoidal abelian category; this is an interesting weak form of the generalised Deligne hypothesis.
	www.ics.mq.edu.au /CoACT/projects (1438 words)

	math lessons - Natural number object (Site not responding. Last check: 2007-10-22)
		In category theory, a natural number object (nno) is an object endowed with a recursive structure similar to natural numbers.
		If a cartesian closed category has weak nnos, then every slice of it also has a weak nno.
		Nnos in CCCs or topoi are sometimes defined in the following equivalent way (due to Lawvere): for every pair of arrows g : A → B and f : B → B, there is a unique h : N × A → B such that the squares in the following diagram commute.
	www.mathdaily.com /lessons/Natural_number_object (375 words)

	Effective Cartesian Closed Categories of Domains (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Again the function space is the crucial construction in order to obtain a cartesian closed category and we therefore concentrate on that.
		We introduce a natural notion of e#ective bifinite domains and show that the category of such is cartesian closed.
		7 Cartesian closed categories of algebraic cpo's (context) - Jung - 1990
	citeseer.ist.psu.edu /hamrin01effective.html (354 words)

	open and closed sets (Site not responding. Last check: 2007-10-22)
		What is the difference between open and closed captioning?...
		5 Closed Sets and Open Sets 5.6 The only way to show directly that a subset X of...
		Automatic detection of open and closed separation and attachment lines...
	www.scienceoxygen.com /math/359.html (242 words)

	DERIVING CATEGORY THEORY FROM TYPE THEORY (Site not responding. Last check: 2007-10-22)
		This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories.
		It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics.
		Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed l-calculus with binary products and a unit type.
	www.doc.ic.ac.uk /deptechrep/old/abs9318.html (193 words)

	Category (mathematics) - (Site not responding. Last check: 2007-10-22)
		A small category is a category in which both ob(C) and hom(C) are actually sets and not proper classes.
		The category Grp consisting of all groups with their group homomorphisms
		The category Ab consisting of all abelian groups with their group homomorphisms
	www.grohol.com /psypsych/Small_category (1282 words)

	Category of abelian groups - (Site not responding. Last check: 2007-10-22)
		In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.
		This functor is faithful, and therefore Ab is a concrete category.
		Ab is not cartesian closed (and therefore also not a topos) since it lacks exponential objects.
	www.grohol.com /psypsych/Category_of_abelian_groups (563 words)

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