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Topic: Adjoint functors

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	Adjoint functors - Wikipedia, the free encyclopedia
		Adjoint functors are one method; the notion of universal properties provides another, essentially equivalent but arguably more concrete approach.
		This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid.
		Similarly, the group ring construction yields a functor from groups to rings, left adjoint to the functor that assigns to a given ring its group of units.
	en.wikipedia.org /wiki/Adjoint_functors (3458 words)

	PlanetMath: adjoint functor
		An adjoint to a functor is in some ways like an inverse (as in the case of an adjoint matrix); often formal properties about a functor lead to formal properties of its adjoint (for example the right adjoint to a left-exact functor takes injectives to injectives).
		This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.
		This is version 11 of adjoint functor, born on 2002-02-25, modified 2005-05-15.
	planetmath.org /encyclopedia/AdjointFunctor.html (186 words)

	Adjoint functors (Site not responding. Last check: 2007-10-20)
		Let G : D → Ab be the functor which assigns to each homomorphism its kernel and let F : Ab → D be the morphism which maps the group A to the homomorphism A → 0.
		Similarly, the group ring construction yields a functor from groupss to rings, left adjoint to the functor that assigns to a given ring its group of units.
		If the functor F : C → D is left adjoint to G : D → C and both C and D are additive categories, then both F and G are additive functors.
	www.bidprobe.com /en/wikipedia/a/ad/adjoint_functors.html (3145 words)

	Functor - Wikpedia (Site not responding. Last check: 2007-10-20)
		Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		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.
		Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Functor (1549 words)

	Adjoint - Wikipedia, the free encyclopedia
		In mathematics, the term adjoint applies in several situations.
		Adjoint curve, in the traditional treatment of coherent duality for a linear system of curves
		For the adjoint of a differential operator with general polynomial coefficients see differential operator
	en.wikipedia.org /wiki/Adjoint (153 words)

	Adjoint functors (Site not responding. Last check: 2007-10-20)
		In category theory, sometimes used as a general theory to discuss the structure concept in mathematics as a whole, the existence of many pairs of adjoint functors is one of the major observations.
		It is probably wrong to say that he promoted the adjoint functor concept in isolation: but it was inherent in the whole approach he took to recognise the role of adjunctions.
		The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous; every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (see limit (category theory)).
	www.theezine.net /a/adjoint-functors.html (2409 words)

	Re: Adjoint Functor Pairs Preserve Limits and Colimits in Categories of Biological Systems, Automata and (M,R)Systems
		Adjoint Functors were in troduced by Daniel Kan in a much cited paper back in 1958.
		In essence, in our model, the pair of adjoint functors between different stages of nuclear transplantation, during development of the organism, preserves limits and colimits that are representing essential functional dynamics in supercategories.
		Therefore, consulting the..."working mathematician" chapter on Adjoint Functors, or any textbook of Category Theory would be important for anyone who wishes to understand in further depth the 'strength and usefulness of adjoint functors' for preserving limits and colimits between certain pairs of categories.
	www.panmere.com /rosen/mhout/msg01642.html (593 words)

	Limit (category theory) (Site not responding. Last check: 2007-10-20)
		The importance of adjoint functors lies in the fact that every functor which has a left adjoint (and therefore is a right adjoint) is continuous.
		A covariant functor that commutes with the construction of colimits is said to be cocontinuous or colimit preserving.
		Every functor which has a right adjoint (and hence is a left adjoint) is cocontinuous.
	www.tocatch.info /en/Limit_(categories).htm (1882 words)

	Commutative Rings - Irving Kaplansky
		The idea of an adjoint functor was formulated by Daniel Kan in 1958.Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations.
		Adjoint functors are one method; the notion of universal properties provide another, essentially equivalent but arguably more concrete approach.Universal properties are also based on category theory.
		Then G is right adjoint to F, which expresses the universal property of kernels, and the co-unit of this adjunction yields the natural embedding of a homomorphisms kernel into the homomorphisms domain.A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints.
	www.vikramasila.org /396489_irving-kaplansky_1114291889commutativeringswherecanibuychristianbookonline.html (3392 words)

	Adjoint (Site not responding. Last check: 2007-10-20)
		the adjoint of an operator (adjoint matrix, adjoint operator; see also self-adjoint, hermitian adjoint), in linear algebra or functional analysis;
		the left adjoint or right adjoint functor in a pair of adjoint functors, in category theory;
		an adjoint curve, in the traditional treatment of coherent duality for a linear system of curves.
	www.bidprobe.com /en/wikipedia/a/ad/adjoint.html (128 words)

	Lambek and Scott: Introduction to higher order categorical logic (Site not responding. Last check: 2007-10-20)
		Equivalently, an equivalence is a pair of functors F:A->B and U:B->A with FU iso to 1_B and UF iso to 1_A.
		A functor U:B->A is called "tripleable" or "monadic" if it is a right adjoint and if the comparison functor is an equivalence of categories (so that up to equivalence B is the Eilenberg-Moore category).
		K is a functor since K(eta_A) = eps'_F'A o F'(eta_A) = 1_A, and K(g*f) = eps'F'A'' o F'(mu_A'' o T(g) o f) = eps'F'A'' o F'U'eps'F'A'' o F'U'F'g o F'f = eps'F'A'' o eps'F'U'F'A'' o F'U'F'g o F'f = eps'F'A'' o F'g o eps'F'A' o F'f = K(g) o K(f).
	www.andrew.cmu.edu /user/cebrown/notes/lambekscott.html (4587 words)

	Citations: Every standard construction is induced by a pair of adjoint functors - Kleisli (ResearchIndex)
		Every standard construction is induced by a pair of adjoint functors.
		The latter is given in its most natural form as a construction on objects introduced as algebraic theories in extension form in [11] and called Kleisli triples in [13] both that description and the Kleisli construction generalize quite naturally to precategories and, in fact, to arbitrary....
		Kleisli, H.: Every standard construction is induced by a pair of adjoint functors.
	citeseer.ist.psu.edu /context/910674/0 (587 words)

	[No title]
		adjoint functors and limits We next recall some general facts about limits and adjoint functors: Let C U* FD be a pair of adjoint* functors (i.e., F is (left) adjoint to U), with the natural a* *djunction isomorphism # : HomC(F D; C) ~= HomD(D; UC).
		Definition 4.11.Let C UFD be adjoint functors, and D a left model category with canonical factorizations (of type) as in Definition 4.2.
		It should perhaps be observed that the situation for an abelian ca* tegory M, in which both left and right derived functors* may be defined, is anomolous: * it arises because M may be viewed either as a category of universal algebras or as * a category of universal coalgebras, over itself.
	hopf.math.purdue.edu /Blanc/Blanc_model.txt (7029 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The category of groups is small-complete and the functor forgetting the group structure is limit-preserving.
		Apparently the analogous criterion for representability of a functor was used in Wiles' proof somewhere.
		They're all variations on a criterion for a category having an initial element: a small-complete category with small hom-sets has an initial object iff there's a small family k_i of objects such that every object has a mapping to one of the k_i.
	www.math.niu.edu /~rusin/known-math/00_incoming/freyd (429 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-10-20)
		The symmetrical presentation of a pair of adjoint functors between two categories will be given in Section 7.2, but this raises logical questions because of the choice of a particular product or whatever within its isomorphism class.
		We shall also show that the naturality conditions on adjoint functors - all too easily dismissed as bureaucracy - are directly related to substitution- and continuation-invariance of the rules of type theory.
		When it becomes apparent that it is an adjoint - frequently to something completely trivial, for example pullback between slices is right adjoint to composition - the general case quickly falls into line (Exercise 7.42).
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s70.html (438 words)

	Adjoint functors (Site not responding. Last check: 2007-10-20)
		If F : Set → Group is the functor assigning to each set X the free group over X, and if G : Group → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G.
		Free rings, free abelian groups, and free modules follow the same pattern.
		be the functor which assigns to every group X the pair (X, X) in the product category Group
	www.sciencedaily.com /encyclopedia/adjoint_functors (3182 words)

	Functor
		Note that contravariant functors reverse the direction of composition.
		Note that one can also define a contravariant functor as a covariant functor on the dual category C
		Universal constructions often give rise to pairs of adjoint functors.
	encyclopedie-en.snyke.com /articles/functor.html (1439 words)

	Limit (category theory) (Site not responding. Last check: 2007-10-20)
		Limits and colimits are defined via [[universal propertyuniversal properties]] and as such provide many examples of adjoint functors.
		If J is a small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category (see category theory) C
		Every functor which has a right adjoint (and is a left adjoint) is cocontinuous.
	www.theezine.net /l/limit-category-theory-.html (701 words)

	Reyes-Reyes - Count nouns, mass nouns and their transformations: a category-theoretic unified semantics (Site not responding. Last check: 2007-10-20)
		We organize the nouns in two categories with adjoint functors between them: the plural, from CNs into MNs, and "portion of" in the opposite direction.
		We interpret these nominal categories into the category of kinds and the category of sup-lattices, respectively, and build adjoint functors between them which interpret the adjoint functors at the nominal level.
		This semantics is applied, among others, to study the 8 syllogisms, already considered in the literature, which result from "Claret is wine, wine is liquid, so claret is liquid", by adding the particle "a" to each noun or keeping it as it is.
	reyes-reyes.com /gonzalo/recent_work/logicalfoundation/Count_nouns,_mass_nouns_and_their_transformations/switchLanguage?set_language=en (252 words)

	Graduiertenkolleg Analytische Topologie und Metageometrie (Site not responding. Last check: 2007-10-20)
		Quillen introduces a sequence of seven categories starting with simply connected pointed spaces localized at Q and ending with simply connected commutative differential (graded) coalgebras over Q. In the sequence of categories there is an adjoint pair of functors relating one category to the next one in the sequence.
		The adjoint functors relating the categories in a sequence respect the weak equivalences and define adjoint functors on the associated homotopy categories.
		On the homotopy categories the adjoint functors are equivalences of categories.
	arachne.uni-muenster.de /math/metageo/vorlesung0203-husemoeller.html (156 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		I am interested in the following situation: a contravariant functor adjoint to its own dual, with the unit and counit being the same morphism, but _not_ an iso.
		At 01:29 PM 2/4/97 -0400, you wrote: >I am interested in the following situation: a contravariant functor >adjoint to its own dual, with the unit and counit being the same >morphism, but _not_ an iso.
		I think it is typical: if we call the functor in question F, and if we write J for the unit object, then we should learn easily that F will just be [(_) -> F(J)], i.e., F(J) itself will serve as your A.
	www.mta.ca /~cat-dist/catlist/1999/adj-dual (239 words)

	week78
		Second of all, the definition of adjoint functor, with hom(Lc,d) and hom(c,Rd) being naturally isomorphic, should remind you of adjoint linear operators on Hilbert spaces.
		for all h in H and k in K. In fact, the whole situation with adjoint functors is a kind of "categorified" version of the situation with adjoint linear operators.
		Most interesting, the definition of adjoint operators requires that and be equal, while the definition of adjoint functors requires only that and be naturally isomorphic.
	math.ucr.edu /home/baez/week78.html (2051 words)

	Adjoint functors - Wikpedia (Site not responding. Last check: 2007-10-20)
		A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism
		× D → Set, and the "naturality" requirement is just equivalent to requiring that these two functors be naturally isomorphic.
		If F : Set">Set → Grp">Grp is the functor assigning to each set X the free group over X, and if G : Grp → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Adjoint_functors (3433 words)

	Categories for the Working Mathematician (Graduate Texts in Mathematics) by Saunders Mac Lane [ISBN: 0387984038] - Find ...
		The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits.
		These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors.
		If you already have had experience with such structures, and are simply looking to understand them from a different perspective - this is the book for you.
	www.gettextbooks.com /isbn_0387984038.html (828 words)

	physics - Adjoint
		the adjoint or conjugate transpose of a matrix in linear algebra;
		the left adjoint or right adjoint in a pair of adjoint functors, in category theory;
		the classical adjoint or adjugate of a matrix, related to its inverse.
	www.physicsdaily.com /physics/Adjoint (119 words)

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