
	Where results make sense

Topic: Left adjoint

Related Topics

adjoint functors

Hermitian adjoint

self adjoint

self adjoint operator

adjoint operator

right adjoint

adjoint functor theorem order theory

Dirac adjoint

self adjoint extension

adjoint curve

adjoint matrix

Essentially self adjoint bounded operator

In the News (Mon 28 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Adjoint functors - Wikipedia, the free encyclopedia
		The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory.
		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/Left_adjoint (3560 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/Adjoint.html (186 words)

	Adjoint functors (Site not responding. Last check: 2007-11-06)
		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.
		Universal constructions are more general than adjoint functor pairs: as mentioned earlier, a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D.
		If D is complete, then the functors with left adjoints can be characterized by the Freyd Adjoint Functor Theorem : G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object X of C there exists a family of morphisms f
	www.bidprobe.com /en/wikipedia/a/ad/adjoint_functors.html (3239 words)

	Encyclopedia: Adjoint functors (Site not responding. Last check: 2007-11-06)
		Every adjoint pair of functors defines a unit η;, a natural transformation from the functor 1
		If F  : Set →; 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.
		Let F and G be a pair of adjoint functors with unit η; and co-unit ε.
	www.nationmaster.com /encyclopedia/Adjoint-functors (3432 words)

	Limit (category theory) - Wikipedia, the free encyclopedia
		Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors.
		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.
		Every functor which has a right adjoint (and hence is a left adjoint) is cocontinuous.
	www.wikipedia.org /wiki/Colimit (1864 words)

	Complete lattice - Wikipedia, the free encyclopedia
		Furthermore, morphisms that preserve all joins are equivalently characterized as the lower adjoint part of a unique Galois connection.
		These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets.
		Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction.
	www.wikipedia.org /wiki/Complete_lattice (2029 words)

	Adjoint functors - InfoSearchPoint.com (Site not responding. Last check: 2007-11-06)
		In accordance with the thinking of Saunders MacLane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake.
		A pair of adjoint functors between two partially ordered sets is called a Galois connection.
		If D is complete, then the functors with left adjoints can be characterized by the Freyd Adjoint Functor Theorem : G has a left adjoint if and only if it is continuous and for every object X of C there exists a family of morphisms f
	www.infosearchpoint.com /display/Adjoint_functor (2747 words)

	The Support-Operators Method
		The adjoint of an operator varies with the definition of its associated inner products, but is unique for fixed inner products.
		The flux operator is left in the general form of a discrete vector as defined in Step 1.
		A ghost cell is a non-existent mesh cell that represents a continuation of the mesh across the outer mesh boundary.
	www.lanl.gov /Augustus/Morel99a/node2.html (2954 words)

	Isomorphisms between left and right adjoints, by H. Fausk, P. Hu, and J.P. May (Site not responding. Last check: 2007-11-06)
		There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate ``dualizing object''.
		Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms.
		The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory.
	www.math.uiuc.edu /K-theory/0573 (169 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS H. Abstract.There are many contexts in algebraic geometry, algebraic topol- ogy, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a s* hift of the left* adjoint specified by an appropriate üd alizing object".
		Thus, in the Grothendieck context, the strong symmetric monoidal functor f* is the left adjoint of a left adjoint.
		We ge* n- erally have much better understanding of left* adjoints, so that the compactness criterion is verifiable, but it is the preservation of coproducts by right adjo* *ints that is required in all of the formal proofs.
	hopf.math.purdue.edu /Fausk-Hu-May/FormalFeb16.txt (6914 words)

	Adjoint (Site not responding. Last check: 2007-11-06)
		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 (141 words)

	Domain theory
		Scott noticed in 1969 that in the category of complete lattices and maps preserving directed joins, limit of a sequence of projections (maps with preinverse left adjoints) is isomorphic to the colimit of those left adjoints (called embeddings).
		The purpose of this paper is to express and prove the most general form of the result, which is for filtered diagrams of adjoint pairs between categories with filtered colimits.
		and find that it consists of the projections (continuous surjections with left adjoint) already known to be of importance in the solution of recursive domain equations.
	www.cs.man.ac.uk /~pt/domains (1068 words)

	APPENDIX D
		The conjugations from the pseudounitary group and algebra to the adjoint and coadjoint groups and algebras are given by the square root transformations given in [definition D.3].
		From proposition D.2, there is a non-singular map from Euclidean Hermitean operators to (left) right G(n)-Hermitean operators, and similarly from unitary operators to right (left) G(n)-pseudounitary operators.
		It is a matter of the adjoint action of a Lie group on its Lie algebra.
	graham.main.nc.us /~bhammel/FCCR/apdxD.html (2914 words)

	Adjoint functors - ArtPolitic Encyclopedia of Politics : Information Portal
		All such pairs of adjoint functors arise from universal constructions.
		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)).
		If D is complete (see limit (category theory)), then the functors with left adjoints can be characterized by the Freyd Adjoint Functor Theorem: G has a left adjoint if and only if it is continuous and for every object x of C there exists a family of morphisms f
	www.artpolitic.org /infopedia/ad/Adjoint_functor.html (981 words)

	Exact functor (Site not responding. Last check: 2007-11-06)
		The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F
		The degree to which a left exact functor fails to be exact can be measured with its right derived functors ; the degree to which a right exact functor fails to be exact can be measured with its left derived functorss.
		Left- and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and G is left exact.
	www.sciencedaily.com /encyclopedia/exact_functor (511 words)

	[No title]
		For an associative ring R, let A be the category of left R-modules, let P be the collection of all summands of free R-modules and let E be the collection of all surjections of R-modules.
		Conversely, suppose that i is a degreewise split monomorphism and the cokernel C of i is P-cofibrant.
		We are concerned with two projective classes on the category A. The first is the categori- cal projective class C whose projectives are summands of free modules, whose exact sequences are the usual exact sequences, and whose epimorphisms are the surjections.
	jdc.math.uwo.ca /papers/relative.txt (10321 words)

	[No title]
		The important point is that left adjoints are unique up to natural equivalence (a natural transformation consisting of isomorphisms): See Corollary IV.1 in Maclane's Categories for the Working Mathematician.
		This is what makes it possible to speak of the functor C as the completion (or closure) operator with respect to the notion of completion embodied by the subcategory C_0.
		A pair of adjoint functors produces a monad (and a comonad), and a monad gives rise to many pairs of adjoint functors.
	www.math.niu.edu /~rusin/known-math/99/closure_op (718 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		There are als* o an obvious pair of adjoint* functors c* : M AE sM : ev0: Here c* (the left-adjoint) associates to X 2 M the discrete simplicial object c* on- sisting of X in every dimension, and ev0 sends Z 2 sM to its zeroth object Z0.
		Definition 2.3.A model category M is left proper if the pushout of a weak equiv- alence along a cofibration is again a weak equivalence.
		By using adjointness, this is equivalent to checking that the maps f j are all trivial cofibrations.
	hopf.math.purdue.edu /Dugger/smod.txt (10897 words)

	\bf RESIDUATED POSETS
		We are therefore justified in calling g the right adjoint of f and in calling f the left adjoint of g.
		Q has a right adjoint if and only if f preserves arbitary joins in P. This property, however, fails to be sufficient if P is not a complete lattice.
		Using an arrow to denote the left (and right) residuals of the multiplication in a commutative, residuated poset is motivated by the traditional notation used to describe an important class of these structures - the so-called Heyting lattices.
	www.mtsu.edu /~jhart/NRESLAT.html (5808 words)

	Development of Virtual Photograp
		Left: Unsegmented 2D CT image of fetus of pregnant woman.
		Left: Adjoint MC simulates radiation transport in “backward” direction with the particles gaining energies.
		Left: 3D image in Ultra3D GUI showing the full-body of a fetus phantom.
	www.rpi.edu /dept/radsafe/public_html/projects.htm (1435 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		K(InjA) be the left adjoint of the inclusion K(InjA)* * !
		A coproduct of injective resolutions is again an injectiv* e resolution, and the left* adjoint I~ preserves all coproducts.
		The left hand square commutes because it is obtai* ned from (6.3) by taking left* adjoints.
	hopf.math.purdue.edu /KrauseH/stable.txt (12051 words)

	Universal property - Enpsychlopedia (Site not responding. Last check: 2007-11-06)
		Let F and G be a pair of adjoint functors with unit η and co-unit ε (see the article on adjoint functors for the definitions).
		Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C (equivalently, every object of D).
		The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.
	www.grohol.com /psypsych/Universal_property (1719 words)

	Practical Foundations of Mathematics
		However, it is rarely stated: since the formula for the adjoint is notationally simple, it is used, embedded in more complicated calculations, without appreciating the significance of the theorem.
		Functions which are symmetrically adjoint on the left preserve least upper bounds and functions which are symmetrically adjoint on the right preserve greatest lower bounds.
		By Corollary 3.6.5, a function can have at most one adjoint on each side, but it can have both of them, and there are strings of any finite length of successively adjoint monotone functions.
	www.geocities.com /yury_bendersky/b/f/s36.html (1271 words)

	Adjunctions Between Categories of Domains (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: In this paper we show that there is no left adjoint to the inclusion functor from the full subcategory C 0 of Scott domains (i.e., consistently complete !0algebraic cpo's) to SFP, the category of SFP-objects and Scott-continuous maps.
		We also show there is no left adjoint to the inclusion functor from C 0 to any larger category of cpo's which contains a simple five-element domain.
		As a corollary, there is no left adjoint to the inclusion functor from C 0 to the category of L-domains.
	citeseer.lcs.mit.edu /83324.html (401 words)

	[No title]
		In this ring, addition is the usual addition of polynomials, while multiplication is defined by Xa = aX + a' for any a in R, and extended to R[X;'] by associativity and distributivity.
		- The functions available are: + - * adjoint apply coeff degree derivation expand iszero lcoeff leftgcd leftgcdex leftlcm leftquo leftrem lmonomial monomial polynomial print reductum rightgcd rightgcdex rightlcm rightquo rightrem sametype set_derivation set_X sympow symprod type X - To define a LODO the function Lodo!/polynomial must be used.
		However, since multiplication is not commutative in this ring, sometimes a prefix left or right is necessary to distinguish between different functions.
	cathode.maths.qmw.ac.uk /Computer_Algebra/REDUCE/OLD/ore_lodo/lodo.txt (1843 words)

Try your search on: Qwika (all wikis)