
	Where results make sense

Topic: Functor categories

Related Topics

category theory

abelian categories

limit category theory

Lists of articles by category

kernel category theory

preadditive categories

product category theory

List of dance style categories

Baire category theorem

dual category theory

additive categories

List of computer and video games by category

In the News (Thu 8 Jan 09)

Brad Pitt

Jennifer Garner

Boston College

Steve Ballmer

Rod Blagojevich

Phil Schiller

Gaza City

Jermain Defoe

Lindsay Lohan

Bill Richardson

	[No title] (Site not responding. Last check: 2007-11-07)
		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.
		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.
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
	www.informationgenius.com /encyclopedia/c/ca/category_theory.html (2864 words)

	Functor category (Site not responding. Last check: 2007-11-07)
		Similar to the previous example, the category of k-linear representations of the group G is the same as the functor category k-Vect
		The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.
		The category Cat of all small categories with functors as morphisms is therefore a cartesian closed category.
	www.sciencedaily.com /encyclopedia/functor_category (1000 words)

	Functor category (Site not responding. Last check: 2007-11-07)
		In category theory, the functor s between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformation s between functors.
		Similar to the previous example, the category of k -linear representations of the group G is the same as the functor category k -Vect
		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.
	www.serebella.com /encyclopedia/article-Functor_category.html (1336 words)

	Functor (Site not responding. Last check: 2007-11-07)
		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.
		Constant functor: A very boring functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X.
		A category with a single object is equivalent to a monoid whose elements are morphisms and whose operation is composition.
	www.yotor.com /wiki/en/fu/Functor.htm (1530 words)

	Kids.net.au - Encyclopedia Category theory - (Site not responding. Last check: 2007-11-07)
		Category theory is half-jokingly known as "abstract nonsense".
		Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories.
		Categories, functors and natural transformations were introduced by Eilenberg and MacLane in 1945.
	www.kids.net.au /encyclopedia-wiki/ca/Category_theory (2107 words)

	TAC abstracs
		We shall solve it in the general case of a category with a terminal object; this indicates that a lifting functor carries an imprint which distinguishes it, and this does not depend, for instance, on having enough pullbacks.
		This approach to partiality seems new in the literature on categories of partial maps since there the central role is played by the Kleisli category on the monad: the category of total maps is only instrumental, and always recognized as definable from that of partial maps (when this comes with sufficient structure).
		Functor categories and partial evaluation We propose a semantics in a suitable functor category for a two-level language in Gomard and Jones (1991).
	www.disi.unige.it /eventsandseminars/tac/abstracts97.html (705 words)

	The Dimensional Ladder
		Functors Definition Examples: a functor from the free category on an object, a morphism, an isomorphism, an endo, an auto Example: a functor from a group G to Set is a set acted on by G, or G-set.
		Example: a functor from a group to Top is a continuous action Example: more generally, a functor from a monoid to C is an action of the monoid on some object of C. example: category of representations of various quivers (free categories on graphs) i.
		Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	MATHS: Category Theory
		Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic.
		A Category is a mixture of an algebra and a directed graph.
		A functor is at another level of abstraction - here we take a set of categories and treat them as objects, and look for a natural, mapping between them that preserves their structure - preserves objects and preserves morphisms between objects.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	Category Theory
		Category theory is a mathematical language which arose in the study of limits for universal coefficient theorems in Cech cohomology by Eilenberg and Mac Lane (1942); so the topic has its origins in some sophisticated topology.
		A FUNCTOR F from a category C to a category D is a map from the set of objects of C to the set of objects of D together with a map from the set Hom(X,Y) for any objects X,Y of C to Hom(F(X),F(Y)).
		The category version of the definition of a group: A group is a category with one object in which all the morphisms are isomorphisms.
	education.wichita.edu /alagic/nextpage/categories.htm (1383 words)

	Categories and functors for the structural-phenomenol,ogical modeling (Site not responding. Last check: 2007-11-07)
		Because in the definition of a category, it is not required that its objects should be sets with elements [11], that is usual mathematical objects, a category with its objects being phenomenological senses is called phenomenological category.
		Although such categories may be considered at a very abstract level, the practice of categories and functors, used mostly in the mathematical domain, has shown, as observed before, that the best and fruitful results may be obtained for particular domains of mathematics (for Abelian groups, topological spaces etc.).
		The condition to be imposed for the morphisms of a phenomenological category is to be phenomenological realizable.
	www.racai.ro /MD-Web/Categories.html (2762 words)

	When projective does not imply flat, and other homological anomalies (Site not responding. Last check: 2007-11-07)
		However, the category $\cal M_G$ of Mackey functors for a compact Lie group $G$ is a category of this type in which projective objects need not be so well-behaved.
		Similar misbehavior occurs in two categories of global Mackey functors which are widely used in the study of classifying spaces of finite groups.
		Given the extent of the homological misbehavior in Mackey functor categories described here, it is reasonable to expect that similar problems occur in other functor categories carrying symmetric monoidal closed structures provided by Day's machinery.
	www.emis.ams.org /journals/TAC/volumes/1999/n9/5-09abs.html (394 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Category theory is a branch of abstract mathematics dealing with the algebra of functions.
		Category theory is "typed"; it requires explicit identification of the domains.
		Category theory is a branch of mathematics that attempts to provide insight into the question "What is the fundamental structure?" This approach requires identifying and manipulating the categories involved.
	xp123.com /wwake/dissertation/model.shtml (3369 words)

	Category theory Details, Meaning Category theory Article and Explanation Guide
		General category theory — an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic — came later; it is now applied throughout mathematics.
		Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph.
		These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies.
	www.e-paranoids.com /c/ca/category_theory.html (2389 words)

	Practical Foundations of Mathematics
		The first task of category theory is an organisational one: after various kinds of objects (types, sets, posets, complete semilattices and dcpos) and maps (terms, relations; partial, total, monotone, continuous and structure- preserving functions; and adjunctions) have been introduced, we were able to put them in a common framework as categories.
		Whereas morphisms of a category are in some sense isolated from one another, functors (like the objects which are their values) have a kind of fluidity between them, given by the morphisms of the target category, which we haven't taken into account.
		Functor categories As we observed in Proposition 4.1.5ff, categories may arise as structures as well as congregations.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s48.html (1934 words)

	[No title]
		That category was introduced by Segal [22], and its homotopy theory was studied by Anderson [1] a* *nd Bousfield and Friedlander [3].
		We assume that D is a category under C and that C is skeletally small in the rest of this section.
		The category of commutative monoids in DT is isomorphic to the category of lax symmetric monoidal functors D -!
	hopf.math.purdue.edu /Mandell-May-Schwede-Shipley/mmss1nov14.txt (7400 words)

	Representing general structures with Chu spaces (Site not responding. Last check: 2007-11-07)
		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.
		Unlike all previous universal categories however, Chu is concretely universal in the sense that the embedding preserves the carrier.
		This is the case both for this representation of objects of an arbitrary small category and for the preceding representation of relational structures (which normally form large categories).
	boole.stanford.edu /parikh/node6.html (806 words)

	Category Theory (Site not responding. Last check: 2007-11-07)
		Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science.
		Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations.
		This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines.
	www.andrew.cmu.edu /course/80-413-713 (149 words)

	Abstract (Site not responding. Last check: 2007-11-07)
		According to Goodwillie's calculus of functors, a functor between categories such as Top (the category of all topological spaces) and Spec (the category of spectra) has approximations which are analogous to the approximations to a function in regular calculus given by its Taylor polynomials.
		The analogy extends even further -- the approximations of a functor are even composed of pieces which look like (c_n x^n)/n!.
		Despite the pleasant analogy to the happily simple world of undergraduate calculus, the approximations to a functor between two categories still turn out to be rather complicated objects.
	www.math.virginia.edu /~zga2m/Walters.html (173 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The important role of category theory in this area is discussed and it is shown how the following selected problems are treated using category theory: First, a unified framework for specification logics, second compositional semantics, third partial algebras and their specification, and fourth specifications and models for concurrent systems.
		For the solution of two of the problems classifying categories are used.
		They allow to present categories of algebras as functor categories and to derive a number of important properties from well known results for functor categories.
	tfs.cs.tu-berlin.de /~mgr/abstracts/apcs98b (122 words)

	Citations: functor categories and block structure - Oles (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		A suitable functor category for modeling Algol like languages and languages with dynamic creation of names is Cpo I, where I the the category of nite cardinals and injective maps,....
		3.2 The static category We de ne b D as the functor category Cpo D op, which is a variant of the more familiar topos of presheaves Set D op.
		and in semantic models of the calculus [18, 6] The only place in the literature where functor categories have been used explicitly to justify higher order syntax is [10] It is, however, fair to say that the possibility of using functor categories for HOAS is part of the folklore.
	sherry.ifi.unizh.ch /context/132050/0 (2523 words)

	Categories
		The example to think of is the category in which the objects are sets and the morphisms are functions.
		Mathematically, if quantization were "natural" it would be a functor from the category whose objects are symplectic manifolds (= phase spaces) and whose morphisms are symplectic maps (= canonical transformations) to the category whose objects are Hilbert spaces and whose morphisms are unitary operators.
		A representation of a group, if we think of a group as a category as Sibley suggests, is just a functor from that category to the category Vect of vector spaces.
	math.ucr.edu /home/baez/categories.html (2546 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The interest and the problem stem from the observation that the nerve functor N induces an equivalence between Cat and the category of simplicial sets, after inverting weak equivalences.
		Golasinski's proposed structure failed to satisfy the factorization axiom, but does lead to a closed model category on the category of pro-objects in Cat, as shown in Golasinski, On closed models on the precategory of small categories and simplicial schemes, Uspekhi Mat.
		Besides an aesthetic objection to Thomason's model category structure, I have a practical one: Thomason uses the second subdivision followed by categorical realization as the functor going from simplicial sets to small categories inducing the equivalence of homotopy categories.
	www.lehigh.edu /~dmd1/sc55 (664 words)

	Derived Categories for Dummies, Part IV \| The String Coffee Table
		For instance this derived categroy of coherent sheaves is equivalent to what is called a triangulated Fukaya categroy and also to (at least for a large number of cases) the derived category of representations of some quiver (which I mentioned already in part III).
		For instance the relation between triangulated Fukaya categories and derived categories of coherent sheaves is related to mirror symmetry.
		These complexes are of course precisely the objects in the derived category of coherent sheaves, as described in part I.
	golem.ph.utexas.edu /string/archives/000538.html (1326 words)

	Aspects Of Fractional Exponent Functors (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes.
		We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.
		Introduction In a cartesian closed category E, some objects A may have the property that they are atoms in the sense that the exponent functor () A : E !
	citeseer.ist.psu.edu /234862.html (393 words)

	Categories for Software Engineering
		In the past ten years, several books have been published on category theory either by computer scientists or having computer scientists as a target audience.
		Part I covers some of the basics of category theory, including the “bare essentials” that are addressed in any book, from graphs to universal constructions and functors.
		Part III offers the chance of seeing category theory at work in a more ambitious project — giving semantics to CommUnity, a prototype language for architectural modelling.
	www.fiadeiro.org /jose/CATBook (379 words)

	Lectures on tensor categories and modular functor (Site not responding. Last check: 2007-11-07)
		These lectures are devoted to the discussion of the relation between tensor categories, modular functor, and 3D topological quantum field thery.
		They were written as a textbook; all the results there are known.
		Chapter 6: Moduli spaces and complex modular functor
	www.math.sunysb.edu /~kirillov/tensor/tensor.html (275 words)

	Publications of Brian J. Day
		[Thesis2] Construction of Biclosed Categories (PhD Thesis, University of New South Wales, 1970).
		Categories in which all strong generators are dense.
		Quantum categories, star autonomy, and quantum groupoids, in "Galois Theory, Hopf Algebras, and Semiabelian Categories",
	www.maths.mq.edu.au /~street/Day.pub.html (303 words)

	Functor Categories and Two-Level Languages - Moggi (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Functor Categories and Two-Level Languages - Moggi (ResearchIndex)
		Abstract: We propose a denotational semantics for the two-level language of [GJ91, Gom92], and prove its correctness w.r.t.
		21 functor categories and block structure (context) - Oles - 1985
	citeseer.ist.psu.edu /moggi98functor.html (495 words)

	Some Isomorphisms Between Functor Categories
		We define some well known isomorphisms between functor categories: between $A^{\mathop{\dot\circlearrowright}(o,m)}$ and $A$, between $C^{\mizleftcart A,B\mizrightcart}$ and ${(C^B)}^A$, and between ${\mizleftcart B,C\mizrightcart}^A$ and $\mizleftcart B^A,C^A\mizrightcart$.
		Unfortunately in this paper "functor" is used in two different meanings, as a lingual function and as a functor between categories.
		\em Categories for the Working Mathematician, volume 5 of \em Graduate Texts in Mathematics.
	www.mizar.org /JFM/Vol4/isocat_2.html (135 words)

Try your search on: Qwika (all wikis)