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Topic: Monad (category theory)

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	Monad (category theory) - Wikipedia, the free encyclopedia
		In category theory, a monad or triple is a type of functor, together with two associated natural transformations.
		Monads are important in the theory of pairs of adjoint functors.
		While monads are quite common, making them explicit is less so (the language belongs to the school of Mac Lane, and has rarely been used in the school of Grothendieck, which prefers to write out monads and comonads longhand).
	en.wikipedia.org /wiki/Monad_(category_theory) (1354 words)

	Monad (Site not responding. Last check: 2007-10-14)
		The monad begat the dyad, which begat the numbers, the numbers begat points, which begat lines, which begat two-dimensional entities, which begat three-dimensional entities, which begat bodies, which begat the four elementss earth, water, fire and air, from which the rest of our world is built up.
		This way of putting it is misleading, however; monads do not interact with each other (are "windowless"), but rather are imbued at creation with all their future experiences in a system of pre-established harmony.
		Within mathematics, specifically category theory, a monad is a type of functor important in the theory of adjoint functors.
	bopedia.com /en/wikipedia/m/mo/monad.html (393 words)

	Monad - Wikipedia, the free encyclopedia
		Monad comes from the Greek word μονάς (from the word μόνος, which means "one", "single", "unique") and may refer to:
		Monad, a symbol of God or totality is known in several philosophical circles
		Monads in functional programming are type constructors that are used in functional programming languages to capture various notions of sequential computation
	en.wikipedia.org /wiki/Monad (203 words)

	Monad - Biocrawler (Site not responding. Last check: 2007-10-14)
		The monad begot the dyad, which begot the numbers, the numbers begat points, which begot lines, which begot two-dimensional entities, which begat three-dimensional entities, which begot bodies, which begot the four elements earth, water, fire and air, from which the rest of our world is built up.
		in category theory, a monad, also known as triple, is a type of functor important in the theory of adjoint functors.
		In pure functional programming languages such as Haskell, monads are used as data types that encapsulate the functional I/O-activity, in such a manner that the side-effects of IO are not allowed to spread out of the part of the program that is not functional (imperative).
	www.biocrawler.com /encyclopedia/Monad (595 words)

	Facts about category theory (Site not responding. Last check: 2007-10-14)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		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.supercrawler.com /Facts/category_theory.html (2896 words)

	PlanetMath: monad (Site not responding. Last check: 2007-10-14)
		As an application, monads have been successfully applied in the field of functional programming.
		Thus a mathematical model of computation such as a monad is needed.
		This is version 8 of monad, born on 2002-02-24, modified 2005-01-25.
	planetmath.org /encyclopedia/LeftAndRightIdentityLawsOfAMonad.html (180 words)

	Part III Category Theory: Synopsis (Site not responding. Last check: 2007-10-14)
		Category theory begins with the observation that the collection of all mathematical structures of a given type, together with all the maps between them, is itself an instance of a nontrivial structure which can be studied in its own right.
		Category theory has had great success in the unification of ideas from different areas of mathematics; it has now become an indispensable tool for anyone doing research in topology, abstract algebra, mathematical logic or theoretical computer science (to name but a few).
		This course aims to give a general introduction to the language of category theory, and should therefore be of interest to a large proportion of pure Part III students.
	www.maths.gla.ac.uk /~tl/categories/synopsis.html (341 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.
		In this talk, we will explain why the ultrafilter monad may be replaced by the filter monad in Barr's result without any loss of information, and how this may be related to the existing theory of lax algebras as presented in [3].
	www.math.yorku.ca /Seminars/category (2145 words)

	de monade The word monad comes from the Greek Greek word...
		The monad begat the "dyad", which begat the numbers, the numbers begat points, which begat lines, which begat two-dimensional entities, which begat three-dimensional entities, which begat bodies, which begat the four element elements earth earth, water water, fire fire and air air, from which the rest of our world is built up.
		The monad was thus a central concept in the cosmology cosmology of the Pythagoreans, who held the belief that the world was - "literally" - built up by numbers.
		in category theory category theory, a "monad", also known as "triple", is a type of functor functor important in the theory of adjoint functors adjoint functors.
	www.biodatabase.de /Monad (524 words)

	Category Theory for Computing Science
		Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
		This book is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
		Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another.
	www.cwru.edu /artsci/math/wells/pub/ctcs.html (1715 words)

	Wadler: Monads
		Moggi's use of monads to factor semantics is used to model the composable continuations of Danvy and Filinski.
		Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.
		Category theorists invented monads in the 1960's to concisely express certain aspects of universal algebra.
	homepages.inf.ed.ac.uk /wadler/topics/monads.html (746 words)

	The Information Flow Framework (IFF)
		The Category Theory Ontology is based upon the namespace for large graphs and graph morphisms that includes horizontal multiplication of graphs and a theory of coherence – a category is a monoid in the monoidal category of large graphs.
		Thus, this category can accommodate the truth concept lattice for a FOL language, whose instance collection is the class of all (model-theoretic) structures for that language, and whose formal concepts have as their intents the (closed) ontologies for that language.
		The theory (intent) of the join or supremum of two concepts is the intersection of the theories (conceptual intents), and the theory (intent) of the meet or infimum of two concepts is the theory of the common models or the closure of the union of the theories (conceptual intents).
	suo.ieee.org /IFF/version/20021205.htm (4405 words)

	monad - FOLDOC Definition (Site not responding. Last check: 2007-10-14)
		/mo'nad/ A technique from category theory which has been adopted as a way of dealing with state in functional programming languages in such a way that the details of the state are hidden or abstracted out of code that merely passes it on unchanged.
		A monad has three components: a means of augmenting an existing type, a means of creating a default value of this new type from a value of the original type, and a replacement for the basic application operator for the old type that works with the new type.
		The alternative to passing state via a monad is to add an extra argument and return value to many functions which have no interest in that state.
	www.nightflight.com /foldoc-bin/foldoc.cgi?monad (476 words)

	monad \| English \| Dictionary & Translation by Babylon
		The word monad comes from the Greek word μονάς (from the word μόνος, which means "one", "single", "unique") and has had many meanings in different contexts in philosophy, mathematics, computing and music:
		The monad begot the dyad, which begot the numbers, the numbers begat points, which begot lines, which begat two-dimensional entities, which begat three-dimensional entities, which begat bodies, which begot the four elements earth, water, fire and air, from which the rest of our world is built up.
		In functional programming languages such as Haskell, monads are data types that encapsulate sequential computation, such as I/O- and state-activity or operations which may fail.
	www.babylon.com /definition/monad?uris=!!ARV6FUJ2JP (474 words)

	Category Theory (M24) (Site not responding. Last check: 2007-10-14)
		Category theory begins with the observation (Eilenberg-Mac Lane 1942) that the collection of all mathematical structures of a given type, together with all the maps between them, is itself an instance of a nontrivial structure which can be studied in its own right.
		In keeping with this idea, the real objects of study are not so much categories themselves as the maps between them--functors, natural transformations and (perhaps most important of all) adjunctions.
		A radical rethink of category theory `from the ground up'; it has little in common with the approach adopted in the course, but is worth reading for its style alone
	www.maths.cam.ac.uk /postgrad/courses/damtp/node32.html (407 words)

	The monad laws (Site not responding. Last check: 2007-10-14)
		Monadic operations must obey a set of laws, known as "the monad axioms".
		While it is not necessary to know category theory to create and use monads, we do need to obey a small bit of mathematical formalism.
		An important class of such monads are ones which have a notion of a zero element and a plus operator.
	www.nomaware.com /monads/html/laws.html (1102 words)

	FMCS'04: Abstracts
		Cockett and Lack's restriction categories provide a general framework for working with abstract categories of partial maps in which the notion of partiality is captured abstractly by a single combinator $\rst{(\;)}$ and four restriction axioms.
		In order to examine categories of partial maps in which not only is the domain of the partial map abstractly defined but also the image of the partial map, in this talk we shall introduce introduce range restriction categories and show their relationship with fibrations.
		The object was to capture categories of partial maps and the approach was to append a new unary operation for the domain of a morphism (as a partial inclusion) with four equations that relate this operation to composition.
	pages.cpsc.ucalgary.ca /~robin/FMCS/FMCS_04/abstracts.html (2416 words)

	Category Theory (M24) (Site not responding. Last check: 2007-10-14)
		Category theory begins with the observation (Eilenberg-MacLane 1942) that the collection of all mathematical structures of a given type, together with all the maps between them, is itself an instance of a nontrivial structure which can be studied in its own right.
		Category theory has had great success in the unification of ideas from different areas of mathematics; it has now become an indispensable tool for anyone doing research in topology, abstract algebra, mathematical logic or theoretical computer science (to name but a few examples).
		This course aims to give a general introduction to category theory, without any (intentional!) bias in the direction of any particular application.
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node26.html (341 words)

	Meningar.com om monad. Monad, info, home mm.
		The fail function is not a technical requirement for inclusion as a monad, but it is often useful in practice and it is included in the Monad class because it is used in Haskell's do-notation...
		The IO Monad Haskell defines a monad called IO that is used to describe computations that interact with the operating system -- in particular to perform input and output...
		Because you can't escape from the IO monad, it is impossible to write a function that does a computation in the IO monad but whose result type does not include the IO type constructor...
	www.meningar.com /monad.html (1529 words)

	Category Theory \| Lambda the Ultimate
		Effects qua theories are then combined by appropriate bifunctors (on the category of theories).
		We give a theory of the commutative combination of effects, which in particular yields Moggi's side-effects monad transformer (an application is the combination of side-effects with nondeterminism).
		And we give a theory for the sum of computational effects, which in particular yields Moggi's exceptions monad transformer (an application is the combination of exceptions with other effects).
	lambda-the-ultimate.org /taxonomy/term/22 (1049 words)

	The Haskell Programmer's Guide to the IO Monad --- Don't Panic \| Lambda the Ultimate
		It seems to provide little intuition or motivation for monads or even why you should care about their theoretical underpinnings nor does it show how monads are used in Haskell have much to do with monads in CT. It's far from alone in the second category.
		One of the things in the former case is in CT one of the main uses of monads is for algebras as one would come across in universal algebra.
		First, looking from the 2-categorical perspective, the level at which monads naturally reside, monads are quite clearly the CT version of closure operators a la Category Theory as Coherently Constructive Lattice Theory.
	lambda-the-ultimate.org /node/view/1183 (1970 words)

	Casual Category Theory - Spring 2000
		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.
		The idea is that as such, some of them will be preserved by adjoints relating T categories of other models (synchronisation trees, event structures, petri nets,...).
	www.brics.dk /~varacca/CCT/cct-spring00.html (504 words)

	[No title]
		This is what makes it possible to speak of the functor C as the completion (or closure) operator with respect to the notion of completion embodied by the subcategory C_0.
		I had to expand my category X' and suitably generalize my operator C so as to achieve a category X with functor C so that C actually took X to X (actually, to the subcategory X_0).
		A pair of adjoint functors produces a monad (and a comonad), and a monad gives rise to many pairs of adjoint functors.
	www.math.niu.edu /~rusin/known-math/99/closure_op (718 words)

	Basic Category Theory (Site not responding. Last check: 2007-10-14)
		This course was given to advanced undergraduate and beginning Ph.D. students in the fall of 1994 in Aarhus, as part of Glynn Winskel's semantics course.
		It is, in the author's view, the very minimum of category theory one needs to know if one is going to use it sensibly.
		These are there to give the reader at least a very rough idea of how the theory ``works''.
	www.brics.dk /LS/95/1/BRICS-LS-95-1/BRICS-LS-95-1.html (141 words)

	Gmane -- Mail To News And Back Again
		Basically, though, the Haskell implementation _is_ the category theoretic definition of monad, with bind/return used instead of (f)map/join/return as described below.
		> > Well, a monad over a category C is an endofunctor T on C, together > with a pair of natural transformations eta: 1 -> T, and mu: T^2 -> T > such that > 1) mu.
		T) = id_C > > In Haskell, a monad is an endofunctor on the category of all Haskell > types and Haskell functions between them.
	article.gmane.org /gmane.comp.lang.haskell.cafe/8046 (336 words)

	Monad - TunesWiki (Site not responding. Last check: 2007-10-14)
		A recent, in-depth, seemingly practical tutorial "All About Monads".
		Monads (thanks to Wayback Machine) ("old link" MIA), a collection of references to literature about monads (mostly computer science) by "Lisa A. Walton" ("old home-page" MIA).
		About optimizing certain kinds of monads, see the work of Jonathan Sobel.
	tunes.org /wiki/Monad?v=1 (56 words)

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