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Topic: Faithful functor

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Functor

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	Springer Online Reference Works
		Two-place functors that are covariant in both arguments are called bifunctors.
		As a rule, a construction that may be defined for any object of a category or for any sequence of objects of a fixed length, independently of the individual properties of the objects, is likely to be functorial.
		is called faithful if these mappings are all injective, and full if they are all surjective.
	eom.springer.de /f/f042140.htm (438 words)

	YourArt.com >> Encyclopedia >> functor (Site not responding. Last check: 2007-11-04)
		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.yourart.com /research/encyclopedia.cgi?subject=/functor (1848 words)

	Forgetful functor - Wikipedia, the free encyclopedia
		The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output.
		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.
		Functors which forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.
	en.wikipedia.org /wiki/Forgetful_functor (734 words)

	Concrete category
		Most categories considered in everyday life are concrete; examples are the category of topological spaces with continuous maps as morphisms or the category of groups with group homomorphisms as morphisms.
		If C is a concrete category, then there exists a forgetful functor F : C → Set which assigns to every object of C the underlying set and to every morphism in C the corresponding function.
		In the formal approach, a concrete category is defined as a category together with a faithful functor into the category of sets.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Concrete_category.html (160 words)

	Faithful Functor -- from Wolfram MathWorld
		A functor is said to be faithful if it is injective on maps.
		category of sets is faithful, but it identifies non-isomorphic groups having the same underlying set.
		A functor which is injective both on objects and maps is sometimes called an embedding.
	mathworld.wolfram.com /FaithfulFunctor.html (131 words)

	The Ultimate Functor Dog Breeds Information Guide and Reference
		That is, functors must preserve identity morphisms and composition of morphism.
		This is an incorrect usage of the prefix "co", which in a categorical context usually means "reverse all arrows".
		Consequently a cofunctor, properly speaking, is the same type of object as a functor.
	www.dogluvers.com /dog_breeds/Functor (1445 words)

	Wikinfo \| Functor (Site not responding. Last check: 2007-11-04)
		F(g ο f) = F(g) ο F(f) for all morphisms f:X → Y and g:Y→ Z. That is, functors must preserve identity morphisms and composition of morphism.
		That is, instead of saying F: C→ D is a contravariant functor, they simply write F: Cop → D (or sometimes F:C → Dop) and call it a functor.
		Dual vectorspace: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.
	www.wikinfo.org /wiki.php?title=Functor (1552 words)

	Re: Extending the n-category table
		>given groupoids c,d and a functor u:c->d, the objects of c can >be thought of via the forgetful functor u as objects of d with >an extra _property_ iff u is full and faithful, as objects of d >with extra _structure_ iff u is faithful, and as objects of d >with extra _stuff_ regardless.
		First note the theorem that a functor between categories is an equivalence iff it's full, faithful, and essentially surjective (that is surjective, not on objects, but on isomorphism classes of objects).
		The functor on the left is also a functor of extra structure, in fact a functor of only extra structure, from the category POV.
	www.lns.cornell.edu /spr/2002-04/msg0041086.html (1355 words)

	Lambek and Scott: Introduction to higher order categorical logic (Site not responding. Last check: 2007-11-04)
		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)

	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.
		This shows that it is misleading to regard forgetful functors as providing a hierarchy of simplicity amongst categories: the notion is entirely dependent upon presentation, and indeed some of the functors in Examples 4.4.4 would be regarded as forgetful by certain authors.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s44.html (1776 words)

	Full and faithful functors - Wikipedia, the free encyclopedia
		a full functor) is a functor which is injective (resp.
		This functor is not full as there are functions between groups which are not group homomorphisms.
		Let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function.
	en.wikipedia.org /wiki/Full_and_faithful_functors (288 words)

	Ccard V2.0 - mantras
		JVO7: A full and faithful functor reflects the property of being a terminal or initial object.
		%% --- JVO7: A faithful functor reflects epis and monos.
		JVO9: An embedding is a functor which is full and faithful and injective on objects.
	www.verify-it.de /sub/ccard/mantras_v20.html (939 words)

	[No title]
		D be an additive functor, and denote by D0 the full subcategory of* * D formed by the objects in the image of F.
		E be an exact functor satisfy* ing Ann F Ann G. Then the composite HE OG with the Yoneda functor* factors through F because F is a cohomological quotient functor.
		Moreover, The* orem 4.4 implies that F is a cohomological quotient functor*.
	www.math.purdue.edu /research/atopology/KrauseH/quotient.txt (17394 words)

	faithful
		Since a person not privy to revelation must either accept it or reject it based solely upon the authority of its proponent, and there is no way for a mere mortal to resolve these conflicting claims by investigation, it is prudent to reserve one's judgment.
		The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form where a, b, c, and d are integers, and ad − bc = 1.
		Through their vocation, the faithful of Opus Dei are equally members of a single portion of the Catholic Church, the "People of God." Opus Dei members,...
	www.experiencefestival.com /faithful (2206 words)

	Categories and Relations - Datamaster User's Manual (Site not responding. Last check: 2007-11-04)
		If ∣A∣ ⊂ Split(E) then given functor T:A→B and for each e ∈ E a choice of splitting in B of T(e), there is a unique functor Split(E) → B that maps the cannonical splitting of e to the chosen splitting.
		A STRONG EQUIVALENCE between A and B is a pair F:A→B, G:B→A such that FG is conjugate to the identity functor on A and GF is conjugate to the identity functor on B.
		A FAITHFUL functor is one that is an embedding and reflects isomorphisms.
	uniquesoftwaredesigns.com /datamaster_demo/doc/user_manual_catrel.html (5463 words)

	[No title]
		Write V for the category of complex vector spaces, write S for the symmetric groupoid (a skeleton of the category of finite sets and permutations), and write C for the cyclic groupoid (whose objects are natural numbers and arrows n --> n are elements of the cyclic group of order n.
		There is a faithful functor J : C --> S which is the identity on objects.
		is faithful (since the inclusion H --> G of any subgroup is a split monic at the H-module level), implies that we obtain a Lie algebra object structure on l in the oplax monoidal category [C,V].
	www.maths.usyd.edu.au /u/AusCat/abstracts/011107rs.html (390 words)

	Representing general structures with Chu spaces (Site not responding. Last check: 2007-11-04)
		We have given two answers to this question elsewhere, one measuring the generality of Chu spaces in terms of arbitrary relational structures and their homomorphisms, of which the foregoing are examples, the other in terms of arbitrary small categories, for which a forgetful functor may or may not be given.
		The advantage of this theorem over the preceding one is that it is concrete with respect to the given forgetful functor, not with respect to one we make up for the occasion.
		It is then straightforward to show that this representation is faithful, full, and concrete.
	boole.stanford.edu /parikh/node6.html (806 words)

	PlanetMath: subcategory
		is called the inclusion functor, or an embedding.
		If it is also full, then we call the corresponding subcategory
		Cross-references: field homomorphism, ring homomorphism, ring, fields, matrix rings, commutative rings, discrete topology, group, topological groups, homomorphisms, additive, abelian groups, Hausdorff spaces, compact, Euclidean spaces, invertible, cardinality, infinite, finite, faithful functor, functor, map, identity, morphisms, objects, subset, collection, category
	planetmath.org /encyclopedia/FullSubcategory.html (242 words)

	Springer Online Reference Works
		is faithful if and only if it is faithful when considered as a functor
		A functor with the latter property is generally called conservative; however, some authors include this condition in the definition of faithfulness.
		In Russian literature there seems to be some confusion between the terms "faithful functor" and "exact functor" , see also Exact functor.
	eom.springer.de /f/f038160.htm (140 words)

	category theory Text - Physics Forums Library
		And the paper you refer to defines star category: it is one equipped with a contrqavariant functor (equivlance I believe as it's invertible) that is the identity on objects and whose square is the identity.
		For him the * is not a functor from Hilb to Hilb.
		Your definition "contravariant idempotent functor that is the identity on objects" doesnt capture one of the essential features of a star category.
	www.physicsforums.com /archive/index.php/t-17484.html (1933 words)

	[No title]
		Since F is an equivale* nce of cate- gories, it is in particular an additive fully faithful* functor.
		This self-equivalence is not isomorphic to the identity functor unless R i* *s free of rank one.
		JP in A. Since the functor A(P, -) is exact, X is the image of th* *e cokernel of f.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	Front: [math.CT/0406615] A Full and faithful Nerve for 2-categories (Site not responding. Last check: 2007-11-04)
		Abstract: We prove that there is a full and faithful nerve functor defined on the category of 2-categories and (normal) lax 2-functors.
		This functor extends the usual notion of nerve of a category and it coincides on objects with the so-called geometric nerve of a 2-category or of a 2-groupoid.
		We also show that (normal) lax 2-natural transformations produce homotopies of a special kind, and that two lax 2-functors from a 2-category to a 2-groupoid have homotopic nerves if and only if there is a lax 2-natural transformation between them.
	front.math.ucdavis.edu /0406.5615v1 (151 words)

	Practical Foundations of Mathematics
		In general, appropriate structure is defined by universal properties, interpretation by a structure-preserving functor from the category of contexts to the semantics.
		Note that we have already used the product functor (Proposition 4.5.13) in this definition, so we would now be stuck if we hadn't insisted on uniqueness of the mediator in the definition of product.
		[[c]] The sorts and operation-symbols of the model are given by the effect of the functor, essentially as in part (a).
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s46.html (1583 words)

	Property, Structure and Stuff
		OTOH, the Abelianization functor Groups -> Abelian groups is surjective on the objects (and on the morphisms for that matter), but groups are not Abelian groups with extra structure, because the functor isn't injective on the morphisms between a given pair.
		First note the theorem that a functor between categories is an equivalence iff it's full, faithful, and essentially surjective (that is surjective, not on objects, but on isomorphism classes of objects).
		In the part of the factorisation of a functor which is forgetting purely structure the fibre is a n.e.
	math.ucr.edu /home/baez/qg-spring2004/discussion.html (10409 words)

	The First Edge of the Cube \| The n-Category Café
		-transport functor is not just locally trivializable – but also that its descent data has extra structure, like topological, or smooth structure.
		Another equivalence which didn’t make it into this paper is that with the notion of smooth functors used by Stolz and Teichner.
		Such arbitrary functors from paths to the group are called generalized connections in parts of the literature.
	golem.ph.utexas.edu /category/2007/05/the_first_edge_of_the_cube.html (3540 words)

	[No title]
		We show the existence of a fully-faithful, exact functor from F to Fquad, which preserves simple objects, where F is the category of functors from the category of finite dimensional vector spaces over the field with two elements to the category of all vector spaces.
		We define a subcategory Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully-faithful functor which preserves simple objects.
		http://hopf.math.purdue.edu/cgi-bin/generate?/Vespa/mixtes Generic representations of orthogonal groups: the mixed functors Christine Vespa In previous work, we defined the category of functors Fquad, associated to vector spaces over the field with two elements equipped with a nondegenerate quadratic form.
	www.lehigh.edu /~dmd1/h0207 (536 words)

	functor - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "functor" is defined.
		Functor : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include functor: contravariant functor, covariant functor, derived functor, faithful functor, adjoint functor, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=functor (145 words)

	Doctrines \| The n-Category Café
		By a standard theorem, such a functor is an equivalence of categories, hence does not forget anything.
		2) Some functors may not hit every isomorphism class of objects in the codomain but are still surjective on the remaining 1- and 2-morphisms, hence full and faithful.
		For other CCCs we may still imagine addressing the functor as a quantum theory, though it might be an exotic sort of quantum theory that no physicist has ever dreamed of, I guess.
	golem.ph.utexas.edu /category/2006/09/doctrines.html (7626 words)

	Basic Properties of Functor Structures
		We start with some basic lemmata concerning the composition of functor structures.
		Later we show two theorems concerning the properties 'full' and 'faithful' of functor structures which are equivalent to the 'onto' and 'one-to-one' properties of their morphmaps, respectively.
		Furthermore, we prove some theorems about the inversion of functor structures.
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol8/functor1.html (163 words)

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