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Topic: Contravariant functors

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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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	PlanetPhysics: covariance and contravariance
		Contravariance is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds.
		The distinction between homology theory and cohomology theory in topology is that homology is a covariant functor, while cohomology is a contravariant functor (it was suggested in a book, Hilton and Wylie, that contrahomology was therefore a better term for cohomology, but this did not catch on).
		By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.
	planetphysics.org /encyclopedia/CovarianceAndContravariance.html (1326 words)

	Functor - Wikipedia, the free encyclopedia
		Diagonal functor: The diagonal functor is defined as the functor from D to the functor category D
		Dual vector space: 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.
		Forgetful functors: The functor U : Grp → 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 (1810 words)

	Covariance and contravariance - Wikipedia, the free encyclopedia
		If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts.
		Contravariant is a mathematical term with a precise definition in tensor analysis.
		The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the tangent bundle as well as the cotangent bundle.
	en.wikipedia.org /wiki/Covariance_and_contravariance (1781 words)

	Category theory (Site not responding. Last check: 2007-10-30)
		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.
	bopedia.com /en/wikipedia/c/ca/category_theory.html (2877 words)

	Equivalence of categories - Wikipedia, the free encyclopedia
		An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor.
		However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping.
		There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.
	en.wikipedia.org /wiki/Equivalence_of_categories (1502 words)

	Functor category (Site not responding. Last check: 2007-10-30)
		In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors.
		The category of presheaves on a topological space X is a functor category: we turn the topological space in a category C having the open sets in X as objects and a single morphism from U to V iff U is contained in V.
		The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.
	bopedia.com /en/wikipedia/f/fu/functor_category.html (942 words)

	Ebook More Info -Cofunctor - Free For You. (Site not responding. Last check: 2007-10-30)
		Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological space s, and algebraic homomorphism s are associated to continuous function maps.
		Algebra of continuous functions: a contravariant functor from the category of topology (with continuous maps as morphisms) to the category of real associative algebra is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
		Contravariant functors on Open(X) are called presheaf on X. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X. Properties
	en_victoria.park.london.ontario.en.lmoney.org (5127 words)

	PlanetMath: derived functor
		A completely analogous construction can be carried out for right-exact functors and for contravariant functors exact on either side, but it is traditional to only describe one case, as doing the others mostly consists of reversing arrows (and replacing “injective” with projective when appropriate), and the result is that of a left derived functor
		Étale cohomology arises as the right derived functors of the global sections functor on the category of étale sheaves; this example includes as special cases the previous two.
		This is version 17 of derived functor, born on 2003-02-10, modified 2006-05-15.
	planetmath.org /encyclopedia/DerivedFunctor.html (382 words)

	Yoneda lemma
		Generally speaking, the Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (where Set is the category of all sets with functions as morphisms).
		This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring.
		The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category.
	www.ebroadcast.com.au /lookup/encyclopedia/yo/Yoneda_lemma.html (634 words)

	[No title]
		C (Fe; C) of the functor* F# are the adjoints of the composites Fe ^ D(d; e) ^ C (Fd; C)"^id-!Fd ^ C (Fd; C)-i!C; where " is an evaluation map of the functor F* and i is an evaluation map of the category C.
		The functors P and L in (4.2) are restrictions of those of (4.1), and the funct* *or J is specified in (3.4).
		The functor f* on spectra is def* ined in terms of the functor* f* on prespectra by f* = Lf*` [4, IIx1].
	www.math.purdue.edu /research/atopology/Mandell-May/mmnov14.txt (8488 words)

	Category theory
		A contravariant functor F between categories C and D is a functor that "turns morphisms around"; the quickest way to define it is as a covariant funtor between C
		Forgetful functors: the functor F : Ring -> Ab which maps a ring to its underlying abelian additive group.
		Functors often describe "natural constructions" and natural transformations often describe "natural homomorphisms" between two such constructions.
	ebroadcast.com.au /lookup/encyclopedia/ca/Category_(mathematics).html (2075 words)

	Good Math, Bad Math : The Category Structure for Linear Logic
		A contravariant functor is a functor from the opposite category.
		The cofunctors are functors in the dual category, but with respect to the category that you're looking at, contravariant functors are not functors, in that they do not respect composition rules correctly.
		Covariant functors, aka "normal" functors, are the ones that are "well-behaved" with respect to the morphisms in the category and compositions.
	scienceblogs.com /goodmath/2006/07/the_category_structure_for_lin_1.php (980 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		A contravariant functor is like a covariant functor except that it reverses the directions of all the arrows.
		F (h, g), F is a functor from X and Y to Z which is covariant in the first variable and contravariant in the second.
		The composition of two functors of the same variance is covariant, and the composition of two functors of opposite variance is contravariant.
	math.ucr.edu /~toby/papers/categories.txt (1722 words)

	Functors for Alternative Categories
		An attempt to define the concept of a functor covering both cases (covariant and contravariant) resulted in a structure consisting of two fields: the object map and the morphism map, the first one mapping the Cartesian squares of the set of objects rather than the set of objects.
		Eventually, the morphism map of a functor from $C_1$ into $C_2$ is a transformation from the arrows of the category $C_1$ into the composition of the object map of the functor and the arrows of $C_2$.\par Several kinds of functor structures have been defined: one-to-one, faithful, onto, full and id-preserving.
		We were pressed to split property that the composition be preserved into two: comp-preserving (for covariant functors) and comp-reversing (for contravariant functors).
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol8/functor0.html (338 words)

	CategoryTheory/Functor - The Haskell Wiki (Site not responding. Last check: 2007-10-30)
		The starting point of Category Theory is the premise that every kind of mathematically structured object comes equipped with a notion of "acceptable" transformation or construction, that is, a morphism that preserves the structure of the object.
		The functors described above are covariant functors, meaning they preserve the direction of arrows.
		Two (covariant) functors F: A → B and G: B → C can be composed to create a new functor GF: A → C.
	www.haskell.org /hawiki/CategoryTheory/Functor (1115 words)

	Functor category
		In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors.
		For every object X of C, let Hom(-,X) be the contravariant representable functor from C to Set.
		has all the formal properties of an exponential object; in particular the functors from E × C → D stand in a natural one-to-one correspondence with the functors from E to D
	www.brainyencyclopedia.com /encyclopedia/f/fu/functor_category.html (1036 words)

	Natural transformation (Site not responding. Last check: 2007-10-30)
		And, oh, a natural transformation is not a functor, it is a map of functors, i.e.
		My functor only works with isomorphism (or things with inverses on the correct side), because I need to invert one of the object morphisms defined by the natural transformation to get the functor...
		Cat is the canonical example, whose 1-morphisms are the functors and 2-morphisms are the natural transformations.
	www.physicsforums.com /showthread.php?t=71414 (1421 words)

	[No title]
		The classical functors w* ere defined by Gersten [16] for algebraic K-theory, by Quinn-Ranicki [33] for algebraic L-t heory, and by using Bott periodicity for C-algebras (see [39] for a discussion of Bott pe* riodicity for C-algebras and also the end of this section for a functorial approach).
		Suppose that X is a contravariant functor from C to CW -COMPLEXES, i.* *e.
		E is the universal approximation from the left by a (weakly) F-exc* isive functor* of a (weakly) F-homotopy invariant functor E from G-F-CW -COMPLEXES to SPEC* *TRA.
	hopf.math.purdue.edu /DavisJ-Lueck/assembly.txt (17837 words)

	Contravariant Functors On Finite Sets And Stirling Numbers (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Contravariant Functors On Finite Sets And Stirling Numbers
		We characterize the numerical functions which arise as the cardinalities of contravariant functors on finite sets, as those which have a series expansion in terms of Stirling functions.
		We give a procedure for calculating the coe#cients in such series and a concrete test for determining whether a function is of this type.
	citeseer.ist.psu.edu /269029.html (285 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		Moreover given a morphism $f\colon X'\to X$ in + $\mathcal{C}$ there is an induced functor $F\colon \mathcal{C}/X' \to + \mathcal{C}/X$ obtained by composition with $f$, and $p_X\circ F = + p_{X'}$.
		A contravariant functor $F\colon +\mathcal{C}\to \mathcal{S}$ is a functor $\mathcal{C}^{\text{opp}}\to +\mathcal{S}$.
		It is clear what the functor $p : \mathcal{S} \to \mathcal{C}$ is. The condition -that $F(U)$ is a groupoid for every $U$ garantees that $\mathcal{S}$ is +that $F(U)$ is a groupoid for every $U$ guarantees that $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$.
	math.columbia.edu /algebraic_geometry/stacks-0.2/src/patches/9.patch (1271 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)

	John Isbell's Adequate Subcategories by F. W. Lawvere (Site not responding. Last check: 2007-10-30)
		The dual notion of a co-adequate subcategory C leads to a contravariant representation of a large category that can be described in terms of (a) quantity-types, (b) functions, and (c) algebraic operations on functions.
		The dual of the notion of geometric continuity (that is, a name for naturality of maps of covariant functors instead of contravariant ones) is (d)"algebraic homomorphism".
		These ideas of John Isbell became fused with the conceptions of Kan, Grothendieck, and Yoneda (emerging in the same period 1958-1960), to form a basic method of analyzing and constructing mathematical categories.
	at.yorku.ca /t/o/p/d/65.htm (715 words)

	Topics: Functors (Site not responding. Last check: 2007-10-30)
		Composition: Contravariant functors can be composed, but their composition is a covariant functor, etc.
		Duality: A contravariant functor with an inverse; Every cat is the domain (and the range) of some duality.
		Forgetful: A functor from a category to another whose structure is less rich.
	www.phy.olemiss.edu /~luca/Topics/f/functor.html (230 words)

	Your Site Title Here...
		Abstract: The general question we consider is that of equivalences of categories of functors (also know as categories of diagrams).
		Our primary focus is on contravariant functors taking values in an arbitrary stable modelcategory.
		We describe certain conditions on a pair (A, B) of small categories that ensure that the categories of contravariant diagrams indexed by A and those indexed by B are Quillen equivalent.
	math.aa.psu.edu /~gtseminar/2004/abstract111804.html (148 words)

	Module (mathematics)
		Any ring R can be viewed as a preadditive category with a single object.
		With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups.
		This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.
	www.askfactmaster.com /Module_(mathematics) (1178 words)

	Citebase - Failure of Brown representability in derived categories (Site not responding. Last check: 2007-10-30)
		In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general.
		The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress.
		There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0001056 (172 words)

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