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Topic: Enriched functor

Related Topics

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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	Functor (Site not responding. Last check: 2007-09-19)
		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.tocatch.info /en/Functor.htm (1585 words)

	Functor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-19)
		Functors can be thought of as (additional info and facts about morphism) morphisms in the category of all ((The slender part of the back) small) categories.
		Forgetful functors: The functor U : Grp → Set which maps a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group to its underlying set and a (additional info and facts about group homomorphism) group homomorphism to its underlying function of sets is a functor.
		Another example is the functor Rng → Ab which maps a (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring to its underlying additive (A group that satisfies the commutative law) abelian group.
	www.absoluteastronomy.com /encyclopedia/f/fu/functor.htm (1762 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		globular nerves of simplicially enriched groupoids there is as far as i know essentially just one reasonable concept of the "globular nerve" of a simplicially enriched groupoid.
		when simplicial sets are functorially mapped to bi-pointed simplicially enriched groupoids by the left adjoint f of the functor "hom(v0,v1)", however, the existence of inverses in a simplicially enriched groupoid provides enough flexibility to repair the orientation mismatch problem.
		once we have the desired co-globular object g in the category of simplicially enriched groupoids, we can define the "globular nerve" functor from simplicially enriched groupoids to globular sets as the right adjoint of it's co-continuous extension, or as composing the yoneda embedding with restriction of pre-sheafs along g.
	math.ucr.edu /~jdolan/globular (424 words)

	Functor category (Site not responding. Last check: 2007-09-19)
		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.
		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.
	www.serebella.com /encyclopedia/article-Functor_category.html (1336 words)

	Read about Functor at WorldVillage Encyclopedia. Research Functor and learn about Functor here! (Site not responding. Last check: 2007-09-19)
		Functors can be thought of as morphisms in the category of all (
		smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of
		group to its underlying set and a group homomorphism to its underlying function of sets is a functor.
	encyclopedia.worldvillage.com /s/b/Functor (1248 words)

	Extending Reynolds
		Let P be a model of the polymorphic typed lambda calculus, and let I:P(0)--->E be a fully faithful functor from P(0) into a topos E whose image is closed in E under taking finite limits and exponentials.
		Let C be a ccc with equalizers of all parallel pairs of arrows and suppose that T:C--->C is a C-enriched functor.
		[RP] Reynolds,J.C. and Plotkin,G.D. "On functors expressible in the polymorphic typed lambda calculus".
	www.seas.upenn.edu /~sweirich/types/archive/1988/msg00065.html (827 words)

	Webspace: Chris Holt
		Enriched categories on the other hand are just as nice for us setaholics as for toposophers.
		In Cat, a functor F:C->D is a graph homomorphism (in the intuitive sense of an incidence-preserving map of graphs) which in addition for each edge (x,y) of C specifies a function F_{xy}:C(x,y)->D(x,y) (the map portion of the functor), such that F_{xx}(1_x) = 1_{F(x)} (F preserves identities) and F_{xz}(fg) = F_{xy}(f) F_{yz}(g) (F preserves composition).
		At the heart of the enriched category idea is the notion of an edge-labeled graph of the graph theoretic kind, with at most one edge from x to y.
	homepages.cs.ncl.ac.uk /chris.holt/home.formal/workroom/library/categorical.foundations.html (3049 words)

	This takes back the text of a talk given by Andrée C
		A theorem on extension of an enriched system of structures into an enriched species of structures is given in 1966 (it is used to define the analogue of 'formal derivatives').
		An application of the notion of an enriched species of structures gives a special case of enriched categories: Let C be a category, C°xC acts on the set of morphisms of C (where C° is the opposite category), the corresponding Set-valued functor being Hom.
		To say that the action is enriched in a category H means that the sets Hom(E,E') are 'naturally' equipped with objects of H; this gives a way to add structures on a category, more adapted for 'big' categories.
	www.mcs.le.ac.uk /~ah83/cat-myths/myth0002.html (4088 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.
		Functor Categories We've seen that given categories C and D, there's a set hom(C,D) consisting of functors from C to D. But in fact we can do better: there's a category hom(C,D) whose objects are functors from C to D and whose morphisms are natural transformations between these!
		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.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	[No title]
		Enriched category theory is a well established branch of category theory.
		T* he free category functor* here takes, between any two objects, all strings of composable non-iden* *tity arrows that start at the first object and end at the second.
		Sets is a constant functor with value the * set ob(A) of 4 objects of A. More exactly it is a discrete simplicial set, since all its face *and degeneracy maps are bijections.
	hopf.math.purdue.edu /Porter/s-catsv2.txt (5164 words)

	Mailgate: sci.math.research: Re: Enriched Category Theory: Weighted limit as a right adjo
		a weight F is right adjoint to the >> functor which sends c in C to the functor (j --> Fj.c).
		But there is one point that is however not >clear to me. There is no canonical way to construct the diagonal >functor which would send c in C to the functor (j --> c).
		That's right, in the enriched context, there may not be a diagonal C --> C^J for certain J. This is related to the fact that if 1 denotes the "unit" V-category (i.e.
	mailgate.supereva.it /sci/sci.math.research/msg04416.html (201 words)

	The Analysis of Informatic Phenomena: Research seminars
		My aim is to give a not too technical talk on the subject of categorical structures enriched in a quantaloid.
		I will quickly review the notions of category, functor and distributor enriched in a quantaloid, and indicate how this "holy trinity" gives rise to such things as presheaves, Cauchy completions, and Morita equivalence.
		Finally I want to show how such Q-orders can also be described as categories enriched in the split-idempotent completion of Q: this reconciles the results of Sydneysiders and Lovanists on that subject in the case where Q is a locale.
	web.comlab.ox.ac.uk /oucl/seminars-tt04/extra/stubbe.html (250 words)

	Functor - Wikpedia (Site not responding. Last check: 2007-09-19)
		Note that one can also define a contravariant functor as a covariant functor on the dual category $C^{op} .$
		That is, instead of saying $F: C\rightarrow D is a contravariant$ functor, they simply write $F: C^{op} \rightarrow D (or sometimes F:C \rightarrow D^{op}) and call it a$ functor.
		Power sets: The power set functor P : Set → Set maps each set to its power set and each function $f : X \subseteq Y to the map which sends U \subseteq X to its image f(U) \subseteq Y .$
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Functor (1549 words)

	Dominique Bourn
		SimplE which is a synthetic version of the nerve functor for 2-groupoids and which, allowing to get free from the combinatorial geometric aspect of the pasting used in the case of 2-categories, has a meaning in the purely internal situation.
		E possesses the structure of the iterated T-elements which determines entirely the existence of the nerve functor in the case of the n-groupoids.
		E of pointed objects in E is trivial (each change of base functors is an equivalence), then it is additive (each fibre is additive as well as any change of base functor), led here to some classifying properties for this fibration p.
	www-lma.univ-littoral.fr /~bourn (4038 words)

	Clin2002 Abstract (Site not responding. Last check: 2007-09-19)
		For this purpose I assume a word representation model in which each derived word is seen as the application of a functor morpheme (usually a suffix) to a base morpheme, like in categorial grammar.
		In my model, however, the morphemes are not identified by means of syntactic categories, but by their morphological paradigm, possibly abbreviated by a paradigmatic category.
		Further, I assume that every functor-argument-pair is stored in the lexicon, transparent or not, and that the functor of the next derivation step should be compatible with the paradigm specified by the stored base.
	odur.let.rug.nl /clin2002/Abstracts/koornwinder.html (216 words)

	Functor - Encyclopedia Glossary Meaning Explanation Functor
		Here you will find more informations about Functor.
		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.
		The free functor F : Set → Grp sends every set X to the free group generated by X.
	www.encyclopedia-glossary.com /en/Functor.html (1459 words)

	monoidal enriched natural transformations
		Notice that it is unnecessary to write down a whole bunch of commutative diagrams to define these constructs: the data and axioms inherent in ordinary enriched notions and in braided monoidal notions do the work for you.
		I have a problem which, it seems to me, requires the notion of enriched natural transformation between enriched monoidal functors, but I haven't been able to find a good reference for it.
		I've drawn a couple of commutative diagrams that should appear in the definition of such a construct, but I feel I could be forgetting a dozen more.
	www.forum-one.org /new-5638241-4348.html (1216 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		It sticks in my mind that somewhere (probably Kelly's book on enriched category theory) I have seen the composition of enriched functors expressed in terms of coends.
		More precisely, I recall that it ran like this (and have proved this for ordinary categories): Associate to an enriched functor F:A \rightarrow B the functor Hom(F(-),-):A^{op} \times B \rightarrow V The composition of F and G is then given by \integral^c Hom(F(-),c) \otimes Hom(G(c),-) \cong Hom(G(F(-)),-).
		Now every functor F determines a profunctor F, where F(a,b) is B(Fa,b); and the calculation David refers to (which is a simple application of the Yoneda isomorphism) is just the verification that (GF)* is canonically isomorphic to (G)(F).
	www.mta.ca /~cat-dist/catlist/1999/enriched-functors (536 words)

	Preadditive category - TheBestLinks.com - Preadditive categories, Abelian group, Bilinear operator, Category theory, ... (Site not responding. Last check: 2007-09-19)
		A preadditive category is a category that is enriched over the monoidal category of abelian groups.
		Most functors studied between preadditive categories are additive.
		If C and D are categories and D is preadditive, then the functor category Fun(C,D) is also preadditive, because natural transformations can be added in a natural way.
	www.thebestlinks.com /Preadditive_categories.html (1163 words)

	[No title]
		Especially bearing that in mind, I do not think the technical content of the paper is all that new: the relationships between structural actions and enriched categories have been in the category-theoretic folklore for several decades, hints of it appearing in the various books and papers (generally not referenced by the author).
		A structural functor is a pair consisting of a functor and a natural transformation.
		The formulation of 2.5 states just "a functor equipped..." which leads later to truly ugly formulations like "the functor part of a structural functor" and "transformations that make the functor strustural".
	homepage.mac.com /a.eppendahl/work/papers/strac-reports.txt (1239 words)

	The modular module system
		Functor types are dependent function types: they consist of a module type for the argument, a module type for the result, and a name for the argument, which may appear in the result type.
		For functor applications, we type the functor and its argument, then check that the type of the argument matches the type of the functor parameter.
		As mentioned in section (ref), a difficulty arises when applying a functor with a truly dependent type to a module expression that is not a path.
	pauillac.inria.fr /~xleroy/modmod/modules.html (4452 words)

	Barry Jay's Research Interests: Shape Theory
		For example, Functorial ML supports a class of functors used to support new forms of polymorphism, which go by the name of functorial or shape polymorphism, or polytypy.
		However, such functors need not be shapely over lists, essentially because exponentiating by a fixed object need not be so (unless the object is finite, in the sense of Finite Objects in a Locos.
		Functors, Types and Shapes states the shape approach to polymorphism, and explores its relationship to polytypism, emphasising that shape polymorphic algorithms are parametrically polymorphic.
	www-staff.it.uts.edu.au /~cbj/Publications/shapes.html (1487 words)

	categories: Re: Functorial injective hull.
		Suppose we have a functor E:C --> H together with natural transformation i: Id --> E(C) such that i is monic and,moreover, E is weak left adjoint to the inclusion functor H --> C. Then I claim that under two additional conditions E is genuine adjoint and i is unit of the adjunction.
		The conditions a) and b) are obviously satisfied in the case when E is "injective hull functor" (of coarse a) is true up to iso, again see P,Freyd proof).
		As the unit of the adjunction is monic so the functor E reflects epimorphisms.
	north.ecc.edu /alsani/ct99-00(8-12)/msg00135.html (569 words)

	[No title]
		The use we made of such a V arising from an abelian monoid M was to give an interesting but unusual example of a monoidal functor.
		We observed that a monoidal functor f: V --> Ab was the same thing as an M-algebra, commutative precisely when the monoidal functor f is symmetric.
		Subject: categories: Re: one-object closed categories Concerning categories enriched in monoidal categories with a single object: another example is given by cocycles.
	www.mta.ca /~cat-dist/catlist/1999/comm-monoid (1465 words)

	[No title]
		In the last two sections (6-7) we prove the main theorem: the functor category CK inherits a Thomason model structure, at least when C is right enriched over simplicial sets, and fibrations are preserved by products and inverse limits.
		C be a functor, where C is a complete category equipped with a mapping object functor.
		Proof: The constant functors F (K) = ; and F (K) = * are the initial and termin* *al objects of CK.
	hopf.math.purdue.edu /Weibel/Homotopyends-R.txt (9000 words)

	Overview of my papers
		Mostly for my own convenience, I wrote out the definitions of (classical) bicategory and of functor, transformation and modification between bicategories, together with a bare-bones account of the coherence theorem: Basic bicategories.
		Categories are a special case of generalized multicategories, so the answer to 'what can a generalized multicategory be enriched in?' specializes to tell us what a category can be enriched in.
		The theory of categories enriched in an fc-multicategory is explained as part of the general theory in Generalized enrichment for categories and multicategories, but there are also two direct accounts, not requiring knowledge of generalized multicategories: fc-multicategories, and the more detailed Generalized enrichment of categories.
	www.maths.gla.ac.uk /~tl/overview.html (1703 words)

	DOCUMENTA MATHEMATICA, Vol. 8 (2003), 409-488 (Site not responding. Last check: 2007-09-19)
		In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories.
		This is achieved in a three-step process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors.
		The general setup is shown to describe equivariant stable homotopy theory, and we recover Lydakis' model category of simplicial functors as a special case.
	www.math.uiuc.edu /documenta/vol-08/13.html (123 words)

	[No title]
		These two operations are not in fact iso functors on the whole of “ C but they are on certain convex subcategories, as defined next.
		Definition 3.12 (supportquotienting functor) Let A be a wellsupported precat egory and “ C and C be as in Definition 3.3 and Definition 3.10.
		These graphs are enriched in the next subsection to form raw contexts (graphs with a hole in them), the arrows of the precategory AIxt.
	para.inria.fr /~leifer/articles/leifer-synlt2.txt (12814 words)

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