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Topic: Natural isomorphism

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	Natural transformation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Natural transformations are, after categories and functors, one of the most basic notions of categorical algebra and consequently appear in the majority of its applications.
		These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor.
		The natural transformations from a representable functor to an arbitrary functor F : C → Set are completely known and easy to describe; this is the content of the (Click link for more info and facts about Yoneda lemma) Yoneda lemma.
	www.absoluteastronomy.com /encyclopedia/n/na/natural_transformation.htm (1102 words)

	Science Fair Projects - Natural transformation
		is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors).
		Natural transformations arise frequently in conjunction with adjoint functors.
		The natural transformations from a representable functor to an arbitrary functor F : C → Set are completely known and easy to describe; this is the content of the Yoneda lemma.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Natural_isomorphism (1139 words)

	Isomorphism theorem - Wikipedia, the free encyclopedia
		In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms.
		First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (also called factor groups).
		The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field).
	en.wikipedia.org /wiki/Isomorphism_theorem (334 words)

	PlanetMath: natural transformation (Site not responding. Last check: 2007-11-07)
		is called a natural isomorphism, a natural equivalence, or an isomorphism of functors.
		The composition of two composable functions which are natural transformations is again a natural transformation, and so
		This is version 11 of natural transformation, born on 2002-01-23, modified 2004-03-31.
	planetmath.org /encyclopedia/NaturalTransformation.html (123 words)

	q_duality
		One advantage of the category-theoretic point of view is that it becomes possible to define concretely what it means for isomorphisms to be ``natural''.
		The idea is to look at isomorphisms of whole categories at a time, instead of individual objects; the naturalness consists of compatibility with morphisms between objects.
		There are two concepts: natural isomorphism of two categories, defined with covariant functors, and natural duality, defined with contravariant functors.
	www.math.ucla.edu /~baker/222a/handouts/q_duality/node5.html (107 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Very commonly, certain "natural constructions", such as the fundamental group of a topological space, can be expressed as functors.
		Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another.
		Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach.
	www.informationgenius.com /encyclopedia/c/ca/category_theory.html (2864 words)

	[No title]
		It is natural to ask when OE is an isomorphism in general, and we shall retur* *n to that question in the context of triangulated categories.
		We assume in addition to the isomorphisms ff = ~ffi* , hence ~ßand ~fl, that we are given an object C 2 C together with an isomorphism* (4.1) fS ~=f!C. Observe that the isomorphism* ~flspecializes to an isomorphism (4.2) Df!X ~=f*DX.
		The formal isomorphism theorems We assume throughout this section that C and D are closed symmetric monoidal categories with compatible triangulations and that (f, f) is an adjoint pair * of functors with f strong symmetric monoidal and exact.
	hopf.math.purdue.edu /Fausk-Hu-May/FormalFeb16.txt (6914 words)

	Tensor product - Wikipedia
		Also note that this definition is also naturally associative, and we can use this to define the tensor product for any number of spaces.
		However, linear subspaces of bilinear operators (or in general, multilinear operators) determine natural quotient spaces of the tensor space, which are frequently useful.
		One way of defining the wedge space constructively is by dividing the Tensor space by the subspace generated by all the tensors of n-tuples which are linearily dependant.
	nostalgia.wikipedia.org /wiki/Tensor_product (1090 words)

	Allen (1997), 'The Use of Isomorphism in Modelling the Common Law' [1997] COL 3
		The starting point for isomorphism was the growing realisation in computing science that the costs of software design and maintenance, and the reliability of that software, were improved by structuring the program to reflect the structure of the problem[1].
		The two single largest advantages of using isomorphism in applications are in the savings of time (and thus money) in maintenance, and in providing systems with the ability to verify their results.
		Secondly, it has been argued[9] that using a natural language system enables the systems designer to avoid the "textual baggage" of having separate questions, prompts, and explanations; all of which must be maintained, and checked for accuracy.
	www.austlii.org /au/other/CompLRes/1997/3/1.html (1466 words)

	Natural transformation (Site not responding. Last check: 2007-11-07)
		In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure, i.e.
		The natural transformations from a representable functor to an arbitrary functor F : C
		The context of Mac Lane's remark was the axiomatic theory of homology.
	www.sciencedaily.com /encyclopedia/natural_transformation (919 words)

	eLibrary Project : Isomorphism (Site not responding. Last check: 2007-11-07)
		In mathematics, an "isomorphism" (in Greek language,Greek ''isos'' equal and ''morphe'' shape) is a kind of Map (mathematics),mapping between objects.
		Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures.
		Isomorphic structures are "the same" at some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical.
	elibraryproject.com /info/isomorphism.html (914 words)

	[No title]
		There is a natural isomorphism of commutative k-algebras between the homotopy ring of the commutative Hk-algebra and the homology ring of the corre- sponding E1 differential graded k-algebra.
		The cellular chain complex of GM, CGM, is naturally* a C-algebra, canonically isomorphic to the free C-algebra on CM, CCM. Technically, for a CW Hk-module M, GM is not a CW Hk-module since Hk is not a CW Hk-module [3, III.2.5].
		The following diagram summarizes the natural weak equivalences of Theorem 3.1 for a CW G-algebra A. The solid arrows denote cellular maps of CW G-algebras; the dotted arrow denotes a map of G-algebras that may not be cellular.
	hopf.math.purdue.edu /Mandell/hkalg.txt (12848 words)

	PlanetMath: third isomorphism theorem (Site not responding. Last check: 2007-11-07)
		I think it is not uncommon to see the third and second isomorphism theorems permuted.
		Cross-references: second isomorphism theorems, natural isomorphism, submodules, ideals, normal subgroups, module, ring, group
		This is version 2 of third isomorphism theorem, born on 2001-12-21, modified 2002-08-24.
	planetmath.org /encyclopedia/ThirdIsomorphismTheorem.html (98 words)

	Articles - Natural transformation (Site not responding. Last check: 2007-11-07)
		This "vertical composition" of natural transformation is associative and has an identity, and allows one to consider the collection of all functors C â†’ D itself as a category (see below under Functor categories).
		The natural transformations from a representable functor to an arbitrary functor F : C â†’ Set are completely known and easy to describe; this is the content of the Yoneda lemma.
		What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.
	www.zgrey.com /articles/Naturally_isomorphic (1072 words)

	PlanetMath: natural equivalence (Site not responding. Last check: 2007-11-07)
		is called a natural equivalence (or a natural isomorphism) if there is a natural transformation
		), and composition is defined in the obvious way (for each object compose the morphisms and it's easy to see that this results in a natural transformation).
		This is version 2 of natural equivalence, born on 2002-02-10, modified 2003-07-21.
	planetmath.org /encyclopedia/NaturalEquivalence.html (80 words)

	Natural isomorphism Definition / Natural isomorphism Research (Site not responding. Last check: 2007-11-07)
		In category theory, an abstract branch of mathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; further informally, one might say it is the study of "figures and numbers".
		natural isomorphism is a natural transformation where every t.
		natural isomorphism is not a flip unless the braiding is a symmetry and the monoid is commutative.
	www.elresearch.com /Natural_isomorphism (334 words)

	Practical Foundations of Mathematics
		The naturality condition on the new side of the adjunction defines the effect of the new construction on morphisms.
		The natural bijection is a simple and symmetrical way of presenting an adjunction and remembering (in a subject with plenty of traps for the dyslexic) which is left and which is right.
		Although an ``intuitively'' natural construction usually turns out to be natural in the formal sense, Theorem 7.6.9 shows that naturality may be the point at issue.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s72.html (1307 words)

	The Dimensional Ladder
		Functors Definition Examples: a functor from the free category on an object, a morphism, an isomorphism, an endo, an auto Example: a functor from a group G to Set is a set acted on by G, or G-set.
		Natural transformations between mathematical constructions: Natural isomorphism between identity functor on Vect and double dual.
		Natural transformations as functors F: I x C -> C where I is the free category on a morphism.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	Natural transformation
		is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence).
		Two functors F and G are called naturally isomorphic if there exists a natural isomorphism from F to G.
		Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups isn't complete without a study of homomorphisms, so the study of categories isn't complete without the study of functors.
	news-server.org /n/na/natural_transformation.html (655 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		For finite W a natural transformation from G 2 A to hIW corresponds to an element of IG(W).
		Thus, P is a retract of a wedge of finite spectra, and we have demonstrated that X is the cofibre of a map between A-projective spectra.
		X is a map, then by naturality the restriction of u to W lies in the image of PExt (H-1 W, B), which is trivial because H-1 W is finitely generated.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	Category theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.
		The central concept which is needed for this purpose is called (categorical) limit, and can be dualized to yield the notion of a colimit.
		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.
	encyclopedia.learnthis.info /c/ca/category_theory.html (3218 words)

	Isomorphism
		The idea behind an isomorphism is to realize that two groups are structurally the same even though the names and notation for the elements are different.
		Up to isomorphism, there is only one group with a prime number of elements.
		To show that two groups are not isomorphic, we need to exhibit a structural property of one group not shared by the other.
	www.math.csusb.edu /notes/advanced/algebra/gp/node19.html (393 words)

	Lp space - Wikipedia, the free encyclopedia
		is reflexive for these values of p: the natural monomorphism from L
		is onto, that is, it is an isomorphism of Banach spaces.
		If S is the set of natural numbers, the space
	en.wikipedia.org /wiki/Lp_space (640 words)

	[No title]
		For example, if we have finite sets a and b, and we use a+b to denote their disjoint union, then there is a natural isomorphism between a+b and b+a.
		Then the natural isomorphism between 1+2 and 2+1 can be visualized as the process of passing one dot past two, like this:.
		Following the policy of replacing equations by isomorphisms, let us define the "weak quotient" S//G to be the category with elements of S as its objects, with a morphism g: s -> s' whenever g(s) = s', and with composition of morphisms defined in the obvious way.
	www.math.niu.edu /~rusin/known-math/00_incoming/20 (2649 words)

	Citations: Isomorphism testing for graphs of bounded genus - Miller (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Miller, Isomorphism testing for graphs of bounded genus, Proceedings of the 12th ACM symposium on theory of computing, 1980, pp.
		Thus we obtain a very simple and natural isomorphism algorithm for graphs embeddable in a xed surface; the diculty is only in proving that the algorithm works correctly.
		their stochastic and analogue nature [23] Furthermore, ii) the combinatorial explosion of possible matching solutions is dramatically reduced by the constraint that nodes are distinguishable by their attributes and that we are mainly dealing with planar or other low dimensional graphs (see, e.g.
	sherry.ifi.unizh.ch /context/742623/0 (1056 words)

	Dual space : Duality (linear algebra) (Site not responding. Last check: 2007-11-07)
		As we saw above, if V is finite-dimensional, then V is isomorphic to V, but the isomorphism* isn't natural and depends on the basis of V we started out with.
		There is a natural homomorphism Ψ from V into the double dual V*, defined by (Ψ(v))(φ) = φ(v) for all v in V, φ in V.
		This map Ψ is always injective; it is an isomorphism if and only if V is finite dimensional[?].
	www.eurofreehost.com /du/Duality_(linear_algebra)_3.html (399 words)

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