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Topic: Forgetful functor

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exact functor

functor categories

derived functor

additive functor

representable functor

Tor functors

Ext functors

Enriched functor

contravariant functors

left exact functor

adjoint functor theorem order theory

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	Forgetful functor - Wikipedia, the free encyclopedia
		For example, the forgetful functor from the category of rings to the category of abelian groups assigns to each ring R the underlying additive abelian group of R.
		Concrete categories have forgetful functors to the category of sets -- indeed they may be defined as those categories which admit a faithful functor to that category.
		Forgetful functors tend to have left adjoints which are ' free ' constructions.
	en.wikipedia.org /wiki/Forgetful_functor (278 words)

	Functor - Wikipedia, the free encyclopedia
		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.
		Forgetful functors: The functor U : Grp â’ Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.
	en.wikipedia.org /wiki/Functor (1524 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 - 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/Adjoint_functors (3560 words)

	PlanetMath: forgetful functor
		by ``forgetting'' any imposed mathematical structure is called a forgetful functor.
		Forgetful functors are often instrumental in studying adjoint functors.
		This is version 1 of forgetful functor, born on 2002-05-17.
	planetmath.org /encyclopedia/ForgetfulFunctor.html (107 words)

	Talk:Forgetful functor - Wikipedia, the free encyclopedia
		But a forgetful functor is the second piece of data that comprises the structure of a concrete category, i.e.
		For example, the forgetful functor from the category of rings to the category of abelian groups assigns to each ring R the underlying additive abelian group of R. To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups.
		Since we are used to working with concrete categories concretely, the defining forgetful functor is the "obvious" one.
	en.wikipedia.org /wiki/Talk:Forgetful_functor (303 words)

	Adjoint functors (Site not responding. Last check: 2007-11-06)
		Similarly, the group ring construction yields a functor from group s to rings, left adjoint to the functor that assigns to a given ring its group of unit s.
		Let G be the functor from topological space s to set s that associates to every topological space its underlying set (forgetting the topology, that is).
		Calculus of Functors The 2001 LMS Lectures by Thomas Goodwillie.
	www.serebella.com /encyclopedia/article-Adjoint_functors.html (3353 words)

	Limit (category theory) - Wikipedia, the free encyclopedia
		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.wikipedia.org /wiki/Limit+(categories) (1864 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second.
		Dual vectorspace: an example of a contravariant functor from the category of all real vector spaces to the category of all real vector spaces is given by assigning to every vector space its dual space and to every linear map its dual or transpose.
		Adjoint functors : A functor can be left (or right) adjoint to another functor that maps in the opposite direction.
	www.informationgenius.com /encyclopedia/c/ca/category_theory.html (2985 words)

	Category theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-06)
		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.
		Forgetful functors: the functor F : Ring → Ab which maps a ring to its underlying abelian additive group.
		Functors like these are called representable, and a major goal in many settings is to determine whether a given functor is representable.
	encyclopedia.learnthis.info /c/ca/category_theory.html (3354 words)

	Practical Foundations of Mathematics
		Since the essence of a functor is that it is defined in a ``coherent'' fashion for all objects and morphisms together, the subscripts and superscripts are omitted: we write F X and F f for the application of the functor to an object or morphism.
		The abstract theory of functors is a good example of a unary language (Definition 4.2.5), and would be clearer in the left-to-right notation without operators or brackets.
		A forgetful functor which is replete reflects the means of exchange in the sense that the underlying object may be exchanged for an isomorphic copy and the structure will follow.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s44.html (1854 words)

	Category of topological spaces (Site not responding. Last check: 2007-11-06)
		We have a "forgetful" functor Top → Set which assigns to each topological space the underlying set, and to each continuous map the underlying function.
		This functor is faithful, and therefore Top is a concrete category.
		The forgetful functor has a left adjoint (which equips a given set with the discrete topology) and a right adjoint (which equips a given set with the trivial topology).
	www.wikisearch.net /en/wikipedia/c/ca/category_of_topological_spaces.html (275 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Forgetful and prolongation functors on diagram spectra 12 8.
		Forgetful and prolongation functors on diagram spectra We use the categories DR to reduce comparisons of categories of diagram spect* *ra to comparisons of categories of diagram spaces.
		The canonical functor SF sends n+ to n+ regarded as a discrete based space; it is the restriction to F of the functor SW.
	hopf.math.purdue.edu /Mandell-May-Schwede-Shipley/mmss1nov14.txt (7401 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		C is a functor, then the composite functor A # a ~=iA a oe-!iA X f-!C is an element f(oe) 2 i*AC(a).
		The corresponding functor is specified by the diagrams (d,ffli) 1m _____//CC1n CCC pri
		The colimit functor takes trivial projective cofibrations * i to 45 trivial cofibrations of M; in effect, the colimit functor* is left adjoint to th* e con- stant functor* from A - Set to I-diagrams in A-sets, and the constant functor preserves fibrations and trivial fibrations.
	hopf.math.purdue.edu /Jardine/cat5.txt (13515 words)

	Limit (category theory) (Site not responding. Last check: 2007-11-06)
		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 C
		Ab is the functor which assigns to every J -indexed family of abelian groups its direct product.
		Examples of colimits are given by the dual versions of the ones given above: coproduct s, initial object s, coequalizers, cokernel s, and pushouts, respectively.
	www.serebella.com /encyclopedia/article-Limit_(category_theory).html (2171 words)

	Encyclopedia: Full functor
		In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target.
		For example, let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function.
		Another example is the forgetful functor Ab → Grp.
	www.nationmaster.com /encyclopedia/Full-functor (230 words)

	Representable functor - Enpsychlopedia (Site not responding. Last check: 2007-11-06)
		In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets.
		An arbitrary functor $F:\mathcal C\rightarrow\mathbf{Set} is said to be 'represented by a pair', (A,\phi), where A is an object of \mathcal C and \phi is in F(A), if there is a natural isomorphism \Phi:\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;)\rightarrow F, given by the consistent family of bijections \Phi_X:\mathrm{Hom}_{\mathcal C}(A,X)\rightarrow F(X), such that$
		If $(F(A),\phi) represents this$ functor then $F is the object part of a left-adjoint of G for which the isomorphism \Phi_B is functorial in B and yields the adjointness.$
	www.grohol.com /psypsych/Representable_functor (745 words)

	Forgetful functor Information (Site not responding. Last check: 2007-11-06)
		A forgetful functor is a type of of a given, the functor is simply to take the underlying set of a structure; this is in fact the most common case.
		is then the forgetful functor from \mathcal{C} to \mathbf{Set}, the category of sets.
		For example, the forgetful functor from \mathbf{Mod}(R) (the category of R- module) to \mathbf{Set} has left adjoint F, with X\mapsto F(X), the free R-module with basis X. For a more extensive list, see [Mac Lane].
	www.articleshead.com /show_article/forgetful-functor (315 words)

	Faithful functor - free-definition (Site not responding. Last check: 2007-11-06)
		In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target.
		Note that a faithful functor need not be injective on objects or morphisms.
		For example, the forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms.
	www.free-definition.com /Faithful-functor.html (148 words)

	Functor (Site not responding. Last check: 2007-11-06)
		In category theory a functor is a mapping from one category to another which objects to objects and morphisms to morphisms such a manner that the composition of and the identities are preserved.
		Nowadays functors are used throughout mathematics to relate various categories.
		Introduction to the theory of categories and functors: [by] Ion Bucur and Aristide Deleanu with the collaboration of Peter J. Hilton (Pure and applied mathematics)
	www.freeglossary.com /Functor (207 words)

	Re: Adjoint Functor Pairs Preserve Limits and Colimits in Categories of Biological Systems, Automata and (M,R)Systems
		A pair of Adjoint Functors, F and G is defined between two categories, A and B, F: A---->B and G:B---->A so that they are capable of universal properties such as preserving limits and colimits.
		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 (630 words)

	Practical Foundations of Mathematics
		Show that the functor which assigns the set of components to a graph (Example 7.1.6(c)) preserves finite products but not equalisers or pullbacks.
		Formulate the results of Section 7.6 as a reflection of the 2-category of categories with []-structure as a property and functors preserving it (and natural transformations) into the 2-subcategory where this structure is canonical and preserved on the nose.
		Describe the effect of the functor Q in Notation 7.7.3(b) on morphisms.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s7e.html (1491 words)

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