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Topic: Kernel category theory

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	Kernel (mathematics) - Wikipedia, the free encyclopedia
		Kernels in abstract algebra are general constructions which measure the failure of a homomorphism or function to be injective.
		In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f.
		The kernel pair of a morphism f is defined as a pullback of f with itself.
	en.wikipedia.org /wiki/Kernel_(mathematics) (455 words)

	Kernel (category theory) - Wikipedia, the free encyclopedia
		In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra.
		Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
		That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
	en.wikipedia.org /wiki/Kernel_(category_theory) (836 words)

	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
		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.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	PlanetMath: examples of initial objects and terminal objects and zero objects
		The same is true for the category of abelian groups as well as for the category of modules over a fixed ring.
		In the category of graphs, the null graph is an initial object.
		Similarly, the category of all small categories with functors as morphisms has the empty category as initial object and the one-object-one-morphism category as terminal object.
	planetmath.org /encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html (616 words)

	Adjoint functors (Site not responding. Last check: 2007-10-21)
		(Category theory discusses the structure concept in mathematics as a whole; see also algebraic structure, structure (category theory).) Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians.
		Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations.
		The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification.
	www.bidprobe.com /en/wikipedia/a/ad/adjoint_functors.html (3145 words)

	kernel (mathematics) (Site not responding. Last check: 2007-10-21)
		Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f.
		But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain.
		Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists).
	www.yourencyclopedia.net /Kernel_(mathematics) (242 words)

	Kernel (algebra) : QuicklyFind Info (Site not responding. Last check: 2007-10-21)
		In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
		In this representation, the kernel corresponds to the nullspace of M.
		The categorical generalisation of the kernel as a congruence relation is the kernel pair.
	www.quicklyfind.com /info/Kernel_(algebra).htm (1938 words)

	Kernel (Site not responding. Last check: 2007-10-21)
		The kernel of a seed is all that is within the outer coat of the seed, as the edible substance contained in the shell of a nut; hence, anything included in a shell, husk, or integument.
		The term kernel can also mean a single seed or grain, as "a kernel of corn".
		The kernel of an operating system is its essential component, such as the Linux kernel.
	www.theezine.net /k/kernel.html (138 words)

	Kernel (category theory) (Site not responding. Last check: 2007-10-21)
		Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X which, when composed with f, yields zero.
		In that case, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y.
		To be explicit, if f : X → Y is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of X and the inclusion homomorphism from K to X is a kernel in the categorical sense.
	www.sciencedaily.com /encyclopedia/kernel__category_theory_ (873 words)

	PlanetMath: supplemental axioms for an Abelian category
		The first two are satisfied by definition in an Abelian category, and others may or may not be.
		Every monic is the kernel of its cokernel.
		This is version 7 of supplemental axioms for an Abelian category, born on 2001-12-12, modified 2004-04-07.
	planetmath.org /encyclopedia/Complete8.html (220 words)

	M. SC. THESIS
		In the theory of rings and modules of quotients, there is, associated with each ring of quotients of a ring R, a notion of torsion for modules over R ([Stenström, Rings and modules of quotients, LNM 237, Springer, 1971], II.3).
		The extension of fundamental ideas and constructions of one area of Mathematics to another one is one of the aims of the Theory of Categories.
		At this point it is possible to study torsion theories in categories with initial and terminal objects.
	www.mat.uc.pt /~picado/publicat/Summary.html (1547 words)

	SUO: Composing Ontologies using morphisms and colimits
		Category theory (CT) is a general, powerful, and fundamental approach, which can be used as an alternative to set theory (ST) as a foundation for mathematics in general and the model theories for logic(s) in particular.
		Bottom line: Nothing in the KIF kernel, as it is currently being developed, is incompatible with either a CT or an ST foundation.
		But when extensions to the kernel are being contemplated or developed, we should confer with the CT experts about how such extensions might be defined in a way that does not create incompatibilities with a CT approach.
	grimpeur.tamu.edu /pipermail/kif/2001-January/000319.html (2195 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.
		Category theory shows that in most known algebras and logistic systems, there is a way to construct an equivalent.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	York University: Category seminar
		Under certain hypotheses, this simple remark may be extended to lax algebras, and leads by way of the Kleisli category of the associated monad to a certain "neighborhood presentation" of the theory of lax algebras.
		ABSTRACT: Following the description by Manes [1] of the category of compact Hausdorff spaces as the Eilenberg-Moore category for the ultrafilter monad, Barr [2] showed that by weakening the axioms for a monad and the subsequent algebras, the Eilenberg-Moore category could be seen to be isomorphic to the category of topological spaces.
		Surprisingly, their braided monoidal categories have played a starring role in the recent resurgence of interest n knot theory led by the work of Vaughan Jones.
	www.math.yorku.ca /Seminars/category (2145 words)

	Offers a resource for acne product and more related acne product sites (Site not responding. Last check: 2007-10-21)
		Kernel trick The kernel trick was first published in the paper M. Aizerman, E. Braverman, and L. Rozonoer.
		The kernel trick uses Mercer's theorem, which states that any positive definite kernel K(x, y) can be expressed as a dot product in a high-dimensional space.
		The kernel may be denoted "=f" (or a variation) and may be defined symbolically as Like any equivalence relation, the kernel can be modded out by to form a quotient set.
	discoveryweb.net /acne/acne-product.html (1698 words)

	Kernel - TheBestLinks.com - Cereal, Corn, Linux kernel, Mathematics, ...
		Kernel - TheBestLinks.com - Cereal, Corn, Linux kernel, Mathematics,...
		Kernel, Cereal, Corn, Linux kernel, Mathematics, Operating system, Seed, Kernel...
		This is a disambiguation page, i.e., a navigational aid which lists other pages that might otherwise share the same title.
	www.thebestlinks.com /Kernel.html (212 words)

	Limit (category theory) (Site not responding. Last check: 2007-10-21)
		In category theory, the limit of a functor generalizes the notions of inverse limit and product used in various parts of mathematics.
		If J is a small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category (see category theory) C
		In the category Ab of abelian groups, this for example shows that the kernel of a product of homomorphisms is naturally identified with the product of the kernels.
	www.city-search.org /li/limit-(category-theory).html (945 words)

	Reduced K-theory of Azumaya Algebras, by Roozbeh Hazrat (Site not responding. Last check: 2007-10-21)
		In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors).
		The common examples are kernel or co-kernel of the maps K_i(F) --> K_i(D), where K_i are Quillen K-groups, D is a division algebra and F its centre, or the homotopy fiber arising from the long exact sequence of above map, or reduced Whitehead group SK_1.
		In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it.
	www.math.uiuc.edu /K-theory/0640 (210 words)

	[No title]
		Experimental [Page 3] RFC 2693 SPKI Certificate Theory September 1999 The References section lists all documents referred to in the text as well as readings which might be of interest to anyone reading on this topic.
		The theory was that with enough such signatures, that association could be trusted because not all of these signer would be corrupt.
		Experimental [Page 28] RFC 2693 SPKI Certificate Theory September 1999 If the verifier is operating from an unordered pool of tuples, then a safe rule for name reduction is to apply only those 4-tuples that define a name as a key.
	www.ietf.org /rfc/rfc2693.txt (11728 words)

	The Magma Philosophy
		Magma is a Computer Algebra system designed to solve problems in algebra, number theory, geometry and combinatorics that may involve sophisticated mathematics and which are computationally hard.
		The kernel of Magma contains implementations of many of the important concrete classes of structure in five fundamental branches of algebra, namely group theory, ring theory, field theory, module theory and the theory of algebras.
		Most of the major algorithms currently installed in the Magma kernel are state-of-the-art and give performance similar to, or better than, specialized programs.
	magma.maths.usyd.edu.au /magma/Features/node2.html (357 words)

	internet culture
		One strain of Marxist theory is construed as inherence utopianism.
		One reason to be wary of utopian or dystopian inherence theories is that they encourage a tendency toward blanket denunciation and renunciation of the Internet, or the blanket opposite, when what is needed is a piecemeal evaluation of this or that use of it, this or that tool that is enabled by the Internet meta-tool.
		It would be a logical error, a ``category mistake'' in Rylean terminology, to evaluate the box of paints as a good, poor, or so-so painting.
	www.brandeis.edu /pubs/jove/HTML/V6/iculture.html (8643 words)

	Music and Mathematics
		The theories of semigroups and monoids are not simpler than the theory of groups.
		This book re-invents all music theory on a formal underlying basis of Grothendieck Topologies, which are a kind of topology of sheaves of categories.
		The math involved is primarily group theory, category theory, and topology.
	faculty.washington.edu /jrahn/5752004.htm (5226 words)

	Normal (Site not responding. Last check: 2007-10-21)
		In geometry and physics: a normal is a line perpendicular to a surface.
		In algebra (in particular, group theory): a normal subgroup is a subgroup that is invariant under conjugation.
		In category theory: a normal morphism is a morphism that arises as the kernel or cokernel of some other morphisms.
	www.theezine.net /n/normal.html (330 words)

	kernel - OneLook Dictionary Search
		kernel, kernel, kernel, kernel, kernel, kernel, kernel : PlanetMath Encyclopedia [home, info]
		Phrases that include kernel: apricot kernel oil, peach kernel oil, dirichlet kernel, palm kernel, security kernel, more...
		Words similar to kernel: center, core, essence, gist, heart, inwardness, kerneling, marrow, meat, nitty-gritty, nub, pith, substance, sum, grain, nutmeat, seed, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=kernel (372 words)

	The linux-kernel mailing list FAQ (Site not responding. Last check: 2007-10-21)
		There is a long thread going on about something completely offtopic, unrelated to the kernel, and even some people who are in the "Who's who" section of this FAQ are mingling in it.
		Some kernel developers, including Linus and Marcelo, have chosen to use BitKeeper to manage their kernel source trees, but this does not mean you need to use BitKeeper yourself to maintain your trees or submit patches.
		Also different vendors tend to inject different things into their kernel patch-sets, which again may subtly change data layouts, etc. In stable kernel series great pains are suffered at maintenance so that data layouts of in-kernel APIs (and API calls themselves) are not changed.
	www.debian-ams-sunysb-edu.lkams.kernel.org /pub/linux/docs/lkml (16992 words)

	Functions of Baking Ingredients, NF94-186 (Site not responding. Last check: 2007-10-21)
		Bran particles cut through the gluten during mixing and kneading of bread dough, resulting in a smaller, heavier loaf.
		Wheat germ, though not a flour, is often used in place of part of the flour in recipes for flavor and fiber.
		Protein, vitamins, minerals, and polyunsaturated fats are concentrated in the germ of grain kernels.
	ianrpubs.unl.edu /foods/nf186.htm (2281 words)

	Towards a Unifying Object Role Modelling Theory (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: In this article we briefly present the idea of defining a kernel for object role modelling techniques, upon which different drawing styles can be based.
		We propose such a kernel (the ORM kernel) and define, as a case study, an ER and a NIAM drawing style on top of it.
		One of the prominent advantages of such a kernel is the possibility to build a CASE-tool supporting multiple methods.
	citeseer.ist.psu.edu /697882.html (312 words)

	Directory - CDNet.com (Site not responding. Last check: 2007-10-21)
		The Basic Kernel Source Code Secrets - By William Jolitz and Lynne Jolitz (authors of 386BSD OS).
		LusitanOS - Open source operating system, planned to be an OS entirely made by Portuguese people and built mainly as a way of self-learning and research on operating systems development.
		Operating System Lecture Notes - Notes on operating system theory, source code of describing actual construction of operating system in C programming language.
	www.cdnet.com /cd/index.cgi?dir=/Computers/Programming/Operating_Systems (579 words)

	Paulist Press -- This creative theory about growth and self-empowerment compares a person to a circle that has the ... (Site not responding. Last check: 2007-10-21)
		This creative theory about growth and self-empowerment compares a person to a circle that has the unique ability to keep expanding.
		Her longtime best selling book, The Nibble Theory, is a process for dealing with the world that moves the reader toward personal power and growth arising out of the unique values and strengths of each person.
		Kaleel Jamison (1931—1985) was a pioneer in the field of applied behavioral sciences, focusing on strategic change methodology that used diversity as a catalyst for total organizational improvement.
	www.paulistpress.com /2621-4.html (274 words)

	Comp.lang.modula2: Answers to Common Questions - v1.13 95.05.12 FAQ
		MAS Modula-2 Algebra System From: kredel@unipas.fmi.uni-passau.de (Heinz Kredel) MAS is an experimental computer algebra system combining imperative programming facilities with algebraic specification capabilities for design and study of algebraic algorithms.
		MAS views mathematics in the sense of universal algebra and model theory and is in some parts influenced by category theory.
		PMOS - a multitasking library (ftp.psg.com) The package is potentially of interest to - people who want to write real-time applications for the IBM-PC; - Modula-2 programmers who want a collection of utility modules; - students of operating systems who want to look at real code.
	www.non.com /news.answers/Modula-2-faq.html (1768 words)

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