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Topic: Natural category theory

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	PlanetMath: category theory
		In such a case, the category-theoretic solution is to work in a different category (called an arrow category) where the objects are now morphisms of the original category (in the example of subgroups of a given group, the objects would be injections of a subgroup into a larger group).
		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.
	planetmath.org /encyclopedia/CategoryTheory.html (1637 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".
		Thus, in category theory, the nature of the elements constituting a certain construction is irrelevant.
	www.seop.leeds.ac.uk /archives/fall1997/entries/category-theory (3066 words)

	Natural - Wikipedia, the free encyclopedia
		Natural is defined as "of or relating to nature"; this applies to both definitions of 'nature': 'essence' ("one's true nature") and 'the untouched world' ("force of nature").
		Natural (♮), in music, are the notes A, B, C, D, E, F, and G, as opposed to the sharps/flats.
		Natural transformation, in mathematics, a means of transforming functors in category theory
	en.wikipedia.org /wiki/Natural (410 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".
		Thus, in category theory, the nature of the elements constituting a certain construction is irrelevant.
	www.science.uva.nl /~seop/archives/win2003/entries/category-theory (3074 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, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures.
		From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics.
	plato.stanford.edu /entries/category-theory (11807 words)

	Category theory - FreeEncyclopedia (Site not responding. Last check: 2007-09-21)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		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.
		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 -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-21)
		Category theory is a (Click link for more info and facts about mathematical) mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		Category theory is half-jokingly known as "generalized (Click link for more info and facts about abstract nonsense) abstract nonsense".
		These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for (Click link for more info and facts about constructive mathematics) constructive mathematics.
	www.absoluteastronomy.com /encyclopedia/c/ca/category_theory.htm (3054 words)

	Stephen's Software Blog category theory
		Category theory is a branch of abstract algebra which generalizes both set theory and graph theory.
		Because category theory is a generalization of graph theory, it is natural to ask if it would be useful for working on SCMs, because the version graph is central to understanding SCMs.
		Saunders points out that slice categories (he calls them "comma categories") were long an "secret tool in the arsenal of knowledgeable experts." This motivates me to look at slice categories in the category of versions and patches.
	turnbull.sk.tsukuba.ac.jp /Blogs/Software/CategoryTheory (375 words)

	Good Math, Bad Math : Category Theory: Natural Transformations and Structure
		In my last category theory post, one of the things I mentioned was how category theory lets you explain the idea of symmetry and group actions - which are a kind of structural immunity to transformation, and a definition of transformation - in a much simpler way than it could be talked about without categories.
		A natural transformation is a morphism from functor to functor, which preserves the full structure of morphism composition within the categories mapped by the functors.
		The basic morphisms of a category express the structure of the category; functors express the structure of relationships between categories; and natural transformations express the structure of relationships between relationships.
	scienceblogs.com /goodmath/2006/06/category_theory_natural_transf.php (959 words)

	Category Theory (Reading Course)
		Category theory is a kind of network algebra which provides a general framework for describing mathematical objects and their interrelations.
		A category is a non-null family of arrows with a binary law of composition which is only partially defined; that is, not every pair of arrows is composable.
		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)

	Category Theoretic Perspectives on the Foundations of Mathematics
		Many categories are categories of all examples of a particular kind of mathematical structure (or of all models of a particular theory).
		Thus category theory prefers to avoid talk about elements, and a category theoretic approach to set theory would emphasise talking about the structure of the category of sets through talk about the morphisms within the structure rather than talk about sets and their elements.
		A debate on "set theory versus category theory" (though that characterisation of the debate is also disputed) has raged on the FOM mailing list, and can be observed in the archives.
	www.rbjones.com /rbjpub/philos/maths/faq004.htm (1865 words)

	An Overview of Natural Law Theory
		For Thomas Aquinas, natural law is that part of the eternal law of God ("the reason of divine wisdom") which is knowable by human beings by means of their powers of reason.
		Natural law theory eventually gave rise to a concept of "natural rights." John Locke argued that human beings in the state of nature are free and equal, yet insecure in their freedom.
		Natural law theory is of the "practical order" of things and the first principle of the practical order is a principle that directs human acts in all their operations, and it will be concerned with the "good," since we act in terms of what a least seems good to us.
	radicalacademy.com /philnaturallaw.htm (1901 words)

	Como Category Theory News
		Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size.
		For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness).
		The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it.
	categorytheorynews.blogspot.com (4369 words)

	Category Theory
		Category theory studies structural aspects of mathematics that are common to many fields of mathematics: e.g., algebra, topology, functional analysis, logic, and computer science.
		Category theory is a mathematical language which arose in the study of limits for universal coefficient theorems in Cech cohomology by Eilenberg and Mac Lane (1942); so the topic has its origins in some sophisticated topology.
		The category version of the definition of a group: A group is a category with one object in which all the morphisms are isomorphisms.
	education.wichita.edu /alagic/nextpage/categories.htm (1383 words)

	Monad (category theory) - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-21)
		In category theory, a monad or triple is a type of functor, together with two associated natural transformations.
		Two constructions, the Kleisli category and the category of Eilenberg-Moore algebras, are extremal solutions of the problem of constructing an adjunction that gives rise to a given monad.
		Roughly what goes on is this: while it is simple set theory that a surjective mapping of sets is as good as the imposition of the equivalence relation 'in the same fiber', for geometric morphisms what you should do is pass to such a coalgebra subcategory.
	evil-wire.luvfeed.org /cache/5113 (1070 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 Course 2004
		Examples are the category of rings with ring homomorphisms, and the category of topological spaces with continous maps as morphisms.
		Category theory also provides a formal setting for reasoning about interactions between categories using the concepts of functors and natural transformations.
		However, even though applications of category theory to computer science and logic are covered in the course, a background in computer science is not necessary.
	www.itu.dk /people/mogel/catcourse (454 words)

	Category Theory for Computing Science (Site not responding. Last check: 2007-09-21)
		Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
		This book 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.
	www.case.edu /artsci/math/wells/pub/ctcs.html (1715 words)

	The Phenomenology of Dissipative Structures
		Besides the fact that he suggests unconventional mathematics, category theory, relative to his audience, biologists, one possible reason for the lack of notice is that Rosen does not give a concrete methodology for constructing a theory of "the system" within its context.
		Category theory, or topos theory specifically, are examples of a kind of mathematics that can use abstract forms of entailment that can entail final cause, but divorced from semantics they are just another kind of formalism.
		The idea is to "throw away" the details in terms of "state" and find the correct abstract functions and structures of natural systems depending on the desired level of abstraction and relation to other natural systems, hence their models.
	users.viawest.net /~keirsey/pofdisstruct.html (8904 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)

	Intermediate Depth Representations
		Rosch's classification theory [26] proposes that the most appropriate level of category abstraction for an object is the most cognitively economic one - which she calls the basic level.
		Gluck and Corter [14] propose two metrics for Category Utility which is a context-sensitive measure of the predictive ability of a level of categorization based upon the structure theory.
		Categories which are very accurate but apply to few individuals are not favoured, nor are those that cover a large proportion of the population but because of generality have poor accuracy.
	www.coiera.com /papers/aimj2/aimj.doc.html (6861 words)

	Category Theory
		Category theory is the field of mathematics formalizing algebraic properties of collections of transformations between mathematical objects.
		Formally, a category is a collection of objects, C, such that for each pair of objects (X, Y) in C, there is a collection of morphisms called M(X,Y).
		If that's sounds vague, it is, because category theory abstracts to the point where it doesn't matter very much what kind of stuff nor of arrows are under discussion.
	c2.com /cgi/wiki?CategoryTheory (700 words)

	Category Theory (M24) (Site not responding. Last check: 2007-09-21)
		Category theory begins with the observation (Eilenberg-MacLane 1942) that the collection of all mathematical structures of a given type, together with all the maps between them, is itself an instance of a nontrivial structure which can be studied in its own right.
		In keeping with this idea, the real objects of study are not so much categories themselves as the maps between them--functors, natural transformations and (perhaps most important of all) adjunctions.
		Category theory has had great success in the unification of ideas from different areas of mathematics; it has now become an indispensable tool for anyone doing research in topology, abstract algebra, mathematical logic or theoretical computer science (to name but a few examples).
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node26.html (341 words)

	Category Theory
		Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science.
		Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations.
		This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines.
	www.andrew.cmu.edu /course/80-413-713 (150 words)

	Category Theory for Computing Science (Site not responding. Last check: 2007-09-21)
		Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
		This book is a textbook in basic category theory, written specifically to beread 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.
	www.cwru.edu /artsci/math/wells/pub/ctcs.html (1730 words)

	Categories
		Quantum theory can be thought of as the generalization of classical mechanics you get by dropping the assumption that observable quantities like position and momentum commute.
		Mathematically, if quantization were "natural" it would be a functor from the category whose objects are symplectic manifolds (= phase spaces) and whose morphisms are symplectic maps (= canonical transformations) to the category whose objects are Hilbert spaces and whose morphisms are unitary operators.
		The objects in the category Tang are {0,1,2,...} and the morphisms in Hom(m,n) are (isotopy classes of) tangles with m strands going in and n strands coming out.
	math.ucr.edu /home/baez/categories.html (2616 words)

	Good Math, Bad Math
		This is one of the last posts in my series on category theory; and it's a two parter.
		Category theory provides a good framework for defining linear logic - and for building a Curry-Howard style type system for describing computations with state that evolves over time.
		For me, the frustrating thing about learning category theory was that it seemed to be full of definitions, but that I couldn't see why I should care.
	scienceblogs.com /goodmath/goodmath/category_theory (1469 words)

	Theory - CreationWiki, the encyclopedia of creation science
		A theory is a well-substantiated explanation of some aspect of the natural world.
		In the scientific method, the theory is formed following the testing of a hypothesis and can incorporate facts and scientific law.
		It frequently exists as a belief that can guide behavior, or an organized system of accepted knowledge that applies in a variety of circumstances to explain a specific set of phenomena (i.e.
	creationwiki.org /Theory (96 words)

	The Church Project: Study Group in Category Theory
		It was aimed at providing a insight into category theory for the working programming languages theory inclined person.
		We intended on understanding the theory through examples on how category is applied within the field of programming languages.
		Natural transformations and Yoneda embedding (4.4-4.5 of [BW88])
	types.bu.edu /category.html (187 words)

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