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Topic: Closed monoidal category

	Preadditive category: Definition and Links by Encyclopedian.com - All about Preadditive category
		A preadditive category is a category that is enriched over the monoidal category[?] of abelian groups.
		An additive category is a preadditive category with all finite biproducts.
		A pre-Abelian category is an additive category with all kernels and cokernels.
	www.encyclopedian.com /pr/Preadditive-category.html (1122 words)

	Preadditive category - Wikpedia (Site not responding. Last check: 2007-10-21)
		Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object.
		That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism.
		When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Preadditive_category (1295 words)

	Additive category: Definition and Links by Encyclopedian.com - All about Additive category
		Ab is preadditive because it is a closed monoidal category[?], and the biproduct in Ab is the finite direct sum.
		Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject.
		In fact, it is a theorem that all adjoint functors between additive categories must be additive functors, and most interesting functors studied in all of category theory are adjoints.
	www.encyclopedian.com /ad/Additive-category.html (897 words)

	Cartesian closed category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		These categories are particularly important in (Any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity) mathematical logic and the theory of (Creating a sequence of instructions to enable the computer to do something) programming.
		The category Set of all (A group of things of the same kind that belong together and are so used) sets, with (A mathematical relation such that each element of one set is associated with at least one element of another set) functions as morphisms, is cartesian closed.
		Substitute categories have therefore been considered: the category of (Click link for more info and facts about compactly generated) compactly generated (Click link for more info and facts about Hausdorff space) Hausdorff spaces is cartesian closed, as is the category of Frölicher spaces.
	www.absoluteastronomy.com /encyclopedia/c/ca/cartesian_closed_category.htm (1168 words)

	Pre-Abelian category (Site not responding. Last check: 2007-10-21)
		Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
		Every pre-Abelian category is of course an additive category, and many basic properties of these categories are described under that subject.
		In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic.
	www.sciencedaily.com /encyclopedia/pre_abelian_category (874 words)

	[No title]
		In particular, there is a homotopy category of monoids of topological symmetric spectra, and this homotopy category is equivalent to the homotopy category of monoids of simplicial symmetric spectra.
		Y is an isomorphism in the homotopy category.
		X of monoids is a countable composition of maps Pi -fi!Pi+1, where fi is the pushout in C of a map X^(i+1)^ gi.
	hopf.math.purdue.edu /Hovey/mon-mod.txt (8088 words)

	Preadditive category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A preadditive category is a (A general concept that marks divisions or coordinations in a conceptual scheme) category that is (Click link for more info and facts about enriched) enriched over the (Click link for more info and facts about monoidal category) monoidal category of (A group that satisfies the commutative law) abelian groups.
		In contrast, the category of all ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) groups is not closed.) See (Click link for more info and facts about medial category) medial category.
		If C and D are categories and D is preadditive, then the (Click link for more info and facts about functor category) functor category Fun(C,D) is also preadditive, because (Click link for more info and facts about natural transformation) natural transformations can be added in a natural way.
	www.absoluteastronomy.com /encyclopedia/p/pr/preadditive_category.htm (1551 words)

	Pre-Abelian category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The original example of an additive category is the category Ab of (A group that satisfies the commutative law) Abelian groups.
		For example, in the category of topological Abelian groups, the image of a morphism actually corresponds to the inclusion of the (Termination of operations) closure of the range of the function.
		In many common situations, such as the category of (A group of things of the same kind that belong together and are so used) sets, where images and coimages exist, their objects are (Click link for more info and facts about isomorphic) isomorphic.
	www.absoluteastronomy.com /encyclopedia/p/pr/pre-Abelian_category1.htm (1008 words)

	[No title]
		Has anyone proved that if you take an "algebra" (actually monoid) object in a monoidal biclosed category that has equalizers and coequalizers, then the category of two-sided modules for that algebra is again a monoidal biclosed category.
		In response to Michael Barr's question: Theorem: If V is a closed braided monoidal category which is complete and cocomplete then the bicategory V-Mod of V-categories, V-modules (sometimes called V-bimodules, V-distributors or V-profunctors), and V-module morphisms is a monoidal bicategory (meaning the hom of a tricategory with one object).
		I didn't include the older fact well known to enriched category theorists: even without any symmetry or braiding, V-Mod is a bicategory in which all right extensions and right liftings exist (I don't like the word "biclosed"; I use "left closed", "right closed" and "closed" for both when dealing with a monoidal category).
	www.mta.ca /~cat-dist/catlist/1999/bimod-biclosed (1236 words)

	[No title]
		When C is the stable homotopy category, the cancellation property and the struc- ture of K(C) have been studied extensively by Freyd [12, 13, 14, 15] and Margo* *lis [33].
		C 0be a strong symmetric monoidal functor be- tween closed symmetric monoidal categories with unit objects S and S0.
		Thus we assume that C is the homotopy category HoS obtained by inverting the weak equivalences of a closed symmetric monoidal Quillen model category S.
	hopf.math.purdue.edu /May/PicApril20.txt (6730 words)

	CAKES TALK
		In the second part of the talk I will discuss the definitions of linear function space, tensor and shriek games aiming for a symmetric monoidal closed category of innocent strategies with a !-comonad so that the associated co-Kleisli category is precisely CA.
		The original presentation of a symmetric monoidal closed category of innocent strategies did not allow for a !-comonad.
		In his thesis, Guy McCusker presented two ways of overcoming this difficulty to arrive at the cartesian closed category CA of arenas and innocent strategies which gives rise to a fully abstract model of PCF.
	web.comlab.ox.ac.uk /oucl/seminars-mt00/extra/nickau.html (249 words)

	Symmetric spectra (Site not responding. Last check: 2007-10-21)
		The stable homotopy category of spectra, much studied by algebraic topologists, is a closed symmetric monoidal category (or a category with a tensor product).
		For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category.
		In this paper, we present such a category of spectra: the category of symmetric spectra.
	www.math.uic.edu /~bshipley/symmetric.html (97 words)

	[No title]
		It seems that there are many cases in which this third category is of interest in itself, whether or not one of the two given categories is or is not monadic or comonadic over the other.
		This remark was essentially intended to supply semantically-based examples of closed categories which have one aspect which is linear (in the straightforward sense that coproducts equal products) and an opposite aspect which is cartesian (in the sense that the tensor is the categorical product).
		Rather, within the category whose objects are continuous maps, consider the subcategory wherein the domains of these structural maps are discrete (or zero-dimensional, if that is different in the model of continuity being considered).
	www.mta.ca /~cat-dist/catlist/1999/symmonclocat (1548 words)

	Pre-Abelian category : Pre-abelian category
		A pre-Abelian category is an additive category that has all kernels and cokernels[?].
		Note that thisn'tion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions.
		Thus, the parallel is the inclusion of the range into its closure, which isn't an isomorphism unless the range was already closed.
	www.termsdefined.net /pr/pre-abelian-category.html (935 words)

	Linear Logic and Typed Lambda-Calculus Workshop (Site not responding. Last check: 2007-10-21)
		To support these new definitions, we prove that every symmetric premonoidal category embeds fully into a closed symmetric premonoidal category, and we characterize closed symmetric premonoidal categories in terms of strong monads on the base cartesian category.
		We also show that this structure is category theoretically natural, being given by the category of algebras for a monad on a mild variant of Cat.
		Finally, we show that to give a symmetric premonoidal category is equivalent to giving a fibration with specified structure; and that extra structure on a premonoidal closed category, such as that used to model continuations, transfers elegantly to corresponding structure on the fibration.
	iml.univ-mrs.fr /~ehrhard/Abstracts/Power.html (323 words)

	Additive category (Site not responding. Last check: 2007-10-21)
		In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A
		(Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism composition is bilinear, i.e.
		A pre-Abelian category is an additive category in which every morphism has a kernel and a cokernel.
	www.sciencedaily.com /encyclopedia/additive_category (919 words)

	First Order Linear Logic in Symmetric Monoidal Closed Categories
		That is, the terms of the logic are given by a linear tyoe theory LTT corresponding to the algebraic idea of a symmetric monoidal closed category.
		The study of logic in such categories is motivated by two examples which are derived as linear analogues of presheaf topoi and Heyting valued sets respectively.
		A monoidal factorisation system then gives rise to a structure preserving fibration between symmetric monoidal closed categories, which we term a linear doctrine.
	www.lfcs.informatics.ed.ac.uk /reports/92/ECS-LFCS-92-194 (378 words)

	Binary function
		In category theory, n-ary functions generalise to n-ary morphisms in a multicategory.
		The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category.
		The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.
	www.brainyencyclopedia.com /encyclopedia/b/bi/binary_function.html (698 words)

	CMUC
		For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T,V)-algebra and show that various old and new structures are instances of such algebras.
		Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories.
		As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.
	www.mat.uc.pt /~cmuc/pubdetails.php?pub=661&lid=1 (139 words)

	Casual Category Theory - Spring 2000
		Now we will see the definition of symmetric monoidal category and monoidal closed category V, and the properties of the correspondent V-categories.
		A V-enriched category, in few words, is a category having hom-objects in V, where V is a monoidal category.
		Proofs of some well-known properties of presheave categories, namely: presheave categories are free colimit completions or any presheave functor is isomorphic to a colimit of representables.
	www.brics.dk /~varacca/CCT/cct-spring00.html (504 words)

	Concurrency Abstracts (Site not responding. Last check: 2007-10-21)
		We prove full completeness for a fragment of the linear logic of the self-dual monoidal category of Chu spaces over 2, namely that the proofs between semisimple (conjunctive normal form) formulas of multiplicative linear logic without constants having two occurrences of each variable are in bijection with the dinatural transformations between the corresponding functors.
		The category Chu is concretely universal for much of concrete mathematics; in particular it concretely represents or realizes all categories of relational structures and their homomorphisms, as well as all topological such.
		The category C is realized in Chu(Set,K) where K is the disjoint union of the underlying sets of objects of C. Each object is realized as the normal Chu space (A,X) where X consists of all functions from A in C astricted to K. Chu Spaces and their Interpretation as Concurrent Objects
	boole.stanford.edu /abstracts.html (9620 words)

	Re: category theory <-> lambda calculus?
		Speaking of sloppy: some of the categories I said were Cartesian closed, weren't.
		So Vect is a closed monoidal category w.r.t.
		I guess I made the same dumb mistake for the category of abelian groups.
	www.lns.cornell.edu /spr/2001-03/msg0031712.html (347 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Triangulations, categories and extended topological field theories, by R. Lawrence, to appear in Quantum Topology, eds L. Kauffman and R. Baadtrio, World Scientific, Singapore, 1993.
		Since the 3-dimensional case required the development of new branches of algebra (namely, quantum groups and braided tensor categories), it seems that the higher-dimensional cases will require still further "higher-dimensional algebra." One approach, which is still being born, involves the use of "n-categories," which are generalizations of braided tensor categories suited for higher- dimensional physics.
		Briefly speaking, Gordon-Power-Street use a category they call "Gray," the category of all 2-categories, made into a symmetric monoidal closed category using a modified version of Gray's tensor product.
	math.ucr.edu /home/baez/twf_ascii/week29 (813 words)

	Citations: Autonomous categories - Barr (ResearchIndex)
		Chu s construction takes a closed monoidal category V with pullbacks and completes it to a self dual category Chu(V; k) De Paiva [dP89a, dP89b] and Brown and Gurr [BG90, BGdP91] apply the Chu construction to respectively a version of Godel s Dialectica and Petri nets.
		The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories.
		Most of the book was devoted to the discovery of autonomous categories as full subcategories of seven different categories of uniform or topological algebras over concrete categories that were either equational or reflective....
	citeseer.ist.psu.edu /context/167340/0 (2832 words)

	Preadditive category (Site not responding. Last check: 2007-10-21)
		In mathematics, a preadditive category is a category that is enriched over the monoidal category of abelian groups.
		Any ring, thought of as a category with only one object, is a preadditive category.
		Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.
	www.worldhistory.com /wiki/P/Preadditive-category.htm (1385 words)

	► » Diff as a closed monoidal category? (Site not responding. Last check: 2007-10-21)
		► » Diff as a closed monoidal category?
		Is there any hope that the category of differentiable manifolds and
		Subject: Re: Diff as a closed monoidal category?
	www.science-chat.org /detail-3627457.html (818 words)

	Some Monoidal Closed Categories of Stable Domains and Event Structures (ResearchIndex)
		It results in a monoidal closed category of dI-domains as well as one of stable event structures which can be used to interpret intuitionistic linear logic.
		Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models are discussed.
		2 A monoidal closed category and its relation with categories..
	citeseer.ist.psu.edu /120460.html (434 words)

	Algebras and modules in monoidal model categories (Site not responding. Last check: 2007-10-21)
		One of the main motivations for defining a closed symmetric monoidal category of spectra is to study the associated categories of ring, algebra and module spectra.
		For all of the standard tools of homotopy theory to apply, Quillen model category structures must be constructed on these associated categories.
		In this paper we give general sufficient conditions for producing Quillen model category structures on categories of rings, algebras and modules.
	www.math.uic.edu /~bshipley/ss1.html (110 words)

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