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Topic: Comonad

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	Monad (category theory) - Wikipedia, the free encyclopedia
		The categorical dual definition is a formal definition of a comonad; this can be said quickly in the terms that a comonad for a category C is a monad for the opposite category C
		A comonad in older terminology is a cotriple.
		What was realised in the period 1960 to 1970 is that recognising the categories of coalgebras for a comonad was an important tool of category theory (particularly topos theory).
	en.wikipedia.org /wiki/Monad_(category_theory) (1070 words)

	Beck's monadicity theorem - tScholars.com (Site not responding. Last check: 2007-10-22)
		Its importance arises in relation with descent, in particular in the Grothendieck approach to algebraic geometry.
		Passing to a category of coalgebras for a comonad T is a high-flown way of modelling what taking equivalence classes does, in less touchy situations.
		In 1970 the whole Grothendieck approach via descent data was shown (Benabou and others) to be equivalent, somewhat non-obviously, to the comonad approach.
	www.tscholars.com /encyclopedia/Beck%27s_monadicity_theorem (237 words)

	CoMonad - The Haskell Wiki (Site not responding. Last check: 2007-10-22)
		A comonad is a dual structure to a monad.
		Codata and Comonads in Haskell has a good explanation of the use of comonads, and the information presented here was derived from it.
		The general feel of using a comonad rather than a monad, is that where one might "push" values into a monad to describe an action, one "pulls" contextual values out of a comonad.
	www.haskell.org /hawiki/CoMonad (1662 words)

	[No title]
		Since this map (between rings) already makes R into a -coalgebra when is regarded as a comonad on the category of rings, it also gives rise to* * a -coalgebra structure on the filtered ring R when is regarded as a comonad on the category of filtered rings.
		This natural transforma- tion implies that W is also a comonad on the category of rings, in which the counit is jE and the comultiplication is (E2)-1 tE.
		Moreover, E is an isomorphism of comonads on the category of filtered rings and E(R) is a filtered ring isomorphism.
	hopf.math.purdue.edu /YauD/moduli2.txt (7230 words)

	gmane.comp.lang.haskell.general (Site not responding. Last check: 2007-10-22)
		The central point is that comonads structure the context-dependence in dataflow paradigms in much the same way as monads organize effects.
		Such trees with a distinguished position are of course the zipper datatype and this is a comonad as well.
		And similarly to the comonadic approach to the semantics of dataflow languages one can give a comonadic structure eg to attribute grammar specifications (either purely synthesized attribute grammars or general attribute grammars).
	comments.gmane.org /gmane.comp.lang.haskell.general/12171 (468 words)

	categories: Re: coinduction
		In my opinion, comonads can sometimes play the role of "strengtheners" or "enrichers" for monads, so that while monads can be viewed as a "substrate for structure", comonads are "substrate for semantics".
		It seems that for computer-scientific purposes strength of the monad is too restrictive a requirement, and the author modifies the semantics by assuming strength of the monad onlywith* respecttoacomonad.
		This means that the monad and comonad distribute in such a way that the monad descends to a strong monad on the category of coalgebras over the comonad.
	north.ecc.edu /alsani/ct99-00(8-12)/msg00232.html (577 words)

	Comonads and Haskell (Site not responding. Last check: 2007-10-22)
		Comonads are an abstraction from category theory dualing many qualities of Monads.
		They are conceptually much simpler than arrows but seem to offer a solution to some problems not easily solved by monads.
		The ideas presented here are not novel except for the comonadic combinators for a nicer syntax.
	www.cs.helsinki.fi /u/ekarttun/comonad (203 words)

	Domain Theory for Concurrency (Site not responding. Last check: 2007-10-22)
		Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation.
		Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical constructions in the model.
		The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation to be understood as a parallel composition of independent processes.
	www.brics.dk /RS/03/43/index.html (146 words)

	Special Seminar
		Within the setting of the categorical approach to total functional programming, we introduce a "many-in-one" recursion scheme that neatly unifies a variety of seemingly diverging strengthenings of the basic recursion scheme of iteration.
		The new scheme, termed generalized iteration, is doubly generic: in addition to being parametric in a functor giving rise to an inductive type, it is also parametric in a comonad and a distributive law (of the functor over the comonad) determining a particular recursion scheme for this inductive type.
		Specializations of the scheme for particular comonads and distributive laws include (simple) iteration and mild generalizations of primitive recursion and course-of-value iteration.
	web.comlab.ox.ac.uk /oucl/seminars-mt01/extra/vene.html (113 words)

	The essence of Dataflow Programming by Tarmo Uustalu and Varmo Vene \| Lambda the Ultimate
		This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure context-dependent computation.
		In particular, we develop a generic comonadic interpreter of languages for context-dependent computation and instantiate it for stream-based computation.
		We also discuss distributive laws of a comonad over a monad as a means to structure combinations of effectful and context-dependent computation.
	lambda-the-ultimate.org /node/view/988 (510 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		There are two ways to define a comonad: I.
		extend g) ('fmap' cannot be defaulted, but a comonad which defines 'extend' may simply set 'fmap' equal to 'liftW'.) A comonad providing definitions for 'extend' /and/ 'duplicate', must also satisfy these laws: > extend f == fmap f.
		Converts a list of comonadic functions into a single function -- returning a list of values sequenceW :: Comonad w => [w a -> b] -> w a -> [b] sequenceW [] w = [] sequenceW (f:fs) w = f w : sequenceW fs w
	www.eyrie.org /~zednenem/2004/hsce/Control/Comonad.hs (403 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		The most important example arises from a general construction, a comonad on the category of vector spaces.
		This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps.
		Finally, we derive additional properties of differential categories in certain special cases, especially when the differential comonad is a storage modality, as in linear logic.
	www.math.mcgill.ca /rags/difftl/difftl.abstract.html (191 words)

	Vortrag im Lambda-Kalkül-und-Typen-Club (Site not responding. Last check: 2007-10-22)
		Within the standard categorical approach to programming with total functions, a "many-in-one" recursion scheme is introduced that unifies a variety of seemingly diverging strengthenings of the basic recursion scheme of iteration.
		The new scheme is doubly generic: in addition to being parametric in a functor giving rise to an inductive type, it is also parametric in a comonad and a distributive law (of the functor over the comonad) that together determine a particular recursion scheme for this inductive type.
		Specializations for particular comonads and distributive laws include iteration and (generalizations of) primitive recursion and course-of-value iteration.
	www.tcs.informatik.uni-muenchen.de /~alti/type-club/010518.html (133 words)

	[No title]
		Comonads are related to coinductive data types, such as streams.
		The rest of the paper motivates the use (and definition) of Comonads.
		It seems the essence of the comonad is that they capture a context of context of a computation.
	www.ittc.ku.edu /~kimmell/bib/node1.html (2811 words)

	Stable Bistructure Models of PCF (Site not responding. Last check: 2007-10-22)
		Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions.
		Bistructures form a categorical model of Girard's linear logic consisting of a linear category together with a comonad.
		The comonad has a co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains.
	www.brics.dk /RS/94/13/index.html (152 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		A comonad on an Abelian category which is corresponding to a noncommutative scheme, may be locally (in affine localization charts) compared with comonads arising from coactions of a Hopf algebra if the initial comonad is compatible with the localization in question.
		If, for some localization cover, the local picture is that of a faithfully flat Hopf-Galois extension, then under simple exactness conditions on the cover, there is always a well-defined quotient which is a noncommutative relative scheme in the sense of Rosenberg.
		Formally, the compatibility conditions between the comonads and the localization covers have the formal structure of distributivity laws in category theory, and, in particular, they parallel the picture of entwined structures of Brzezinski and Majid in a related context.
	www.maths.qmw.ac.uk /~majid/Skoabs.html (132 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Although a domain theory for nondeterministic processes it also forms a model of linear logic in which there are comonads interpreting variants of the linear logic exponential.
		The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation which can be understood as a parallel composition of independent processes.
		I'll conclude with a quick discussion of a broader programme of research, towards a comprehensive domain theory for concurrency.
	www-lipn.univ-paris13.fr /equipes/LCR/Seminaire/SEMINAIRE/resume150304.html (146 words)

	The Evolution of a Haskell Programmer
		On 19 June 2001, at the OGI PacSoft Tuesday Morning Seminar Series, Iavor Diatchki presented the paper “Recursion Schemes from Comonads” by Uutsalu, Vene and Pardo [4].
		I attended Iavor’s excellent presentation and remarked that I found the end of the paper rather anti-climactic: after much categorical effort and the definition of several generalized recursion combinators, the main examples were the factorial and Fibonacci functions.
		By the time we arrive at the “pièce de résistance”, the comonadic version of Uutsalu, Vene and Pardo, we have covered most of the underlying ideas and can (hopefully) concentrate better on their specific contributions.
	www.willamette.edu /~fruehr/haskell/evolution.html (2890 words)

	Re: comonads, io
		I'm aware that > the state-monad s->(s,a) has a dual state-in-context > comonad (s,s->a).
		class Comonad w where (=>>) :: w a -> (w a -> b) -> w b (.>>) :: w a -> b -> w b coeval :: w a -> a instance Comonad OI where etc. The first thing I notice is that you can't create objects of type "OI a".
		This would suggest comonads don't need to by type-constructors at all.
	www.mail-archive.com /haskell-cafe@haskell.org/msg02403.html (230 words)

	comon.html (Site not responding. Last check: 2007-10-22)
		A class K of coalgebras for an endofunctor T on the category of sets is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras).
		Then we show that behavioural covarieties K are (isomorphic to) the Eilenberg-Moore categories of coalgebras for certain comonads G^K naturally associated with G^T.
		These are called pure subcomonads of G^T, and a categorical characterization of them is given, involving a pullback condition on the naturality squares of a transformation from G^K to G^T.
	www.mcs.vuw.ac.nz /~rob/papers/comon.html (196 words)

	Control.Comonad
		(fmap cannot be defaulted, but a comonad which defines
		Calls a comonadic function in a modified context
		Converts a list of comonadic functions into a single function returning a list of values
	www.eyrie.org /~zednenem/2004/hsce/Control.Comonad.html (194 words)

	Optimizing Optimal Lambda-Calculus Implementations (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Abstract: In [As94], a correspondence between Lamping-Gonthier's operators for Optimal Reduction of the -calculus [Lam90, GAL92a] and the operations associated with the comonad "!" of Linear Logic was established.
		In this paper, we put this analogy at work, adding new rewriting rules directly suggested by the categorical equations of the comonad.
		These rules produce an impressive improvement of the performance of the reduction system, and provide a first step towards the solution of the well known and...
	citeseer.lcs.mit.edu /asperti94optimizing.html (427 words)

	Premonoidal categories and notions of computation \| Lambda the Ultimate
		And one of the neatest results ever, imo, is that if you take a CCC with a monad and a comonad, you get a categorical model of intuitionistic S4 modal logic.
		This works for nearly any monad and comonad you care to pick.
		Aleks Nanevski's thesis is the place to look -- one of his examples are exceptions, and he makes them symmetric monoidal comonads.
	lambda-the-ultimate.org /node/view/630 (744 words)

	comonads, io (Site not responding. Last check: 2007-10-22)
		I came across an interesting paper "Codata and comonads in Haskell" by Richard Kieburtz, and some other work by Alberto Pardo on comonads.
		Kieburtz proposes an OI comonad, as an alternative to (or something alongside?) the IO monad.
		Are comonads especially appropriate for programming systems that react to changes in their environment?
	haskell.org /pipermail/haskell-cafe/2002-December/003787.html (195 words)

	[No title]
		The appropriate context is the language of comonads and coalgebras over a comonad.
		We describe the structure of the unstable operations on E-cohomology in terms of comonads, in the style of the companion paper on stable operations.
		For practical use, we unpack the comonad information and express it in terms of Hopf rings.
	claude.math.wesleyan.edu /~mhovey/archive/letter05 (1166 words)

	Chu is cofree (Site not responding. Last check: 2007-10-22)
		It induces a comonad on a category of autonomous categories.
		The models of full classical linear logic (with the exponentials) are then obtained as the coalgebras on the models of intuitionistic linear logic --- this time for a comonad derived from the Chu construction.
		In view of the computational interpretations of linear logic, these results suggest an interesting connection of the functional and the concurrent programming.
	www.seas.upenn.edu /~sweirich/types/archive/1993/msg00134.html (143 words)

	BRICS Research Series, Abstracts, 1994
		the comonad used to interpret the ``of course'' modality.
		Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output.
		The comonad of the model has associated co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains.
	www.brics.dk /RS/94/Abs/BRICS-RS-94-Abs/BRICS-RS-94-Abs.html (5995 words)

	Abstract, Test July 2nd 2003 (Site not responding. Last check: 2007-10-22)
		Modal logic features two operators: the monad of modal possibility Diamond, and the comonad of modal necessity Box.
		I will argue that the comonad Box is often a more suitable abstraction for representing effects, and it avoids the restrictions of monadic programming.
		The effects which benefit from Box are those with a delimited scope (i.e., those which admit a notion of handling, like exceptions).
	www.itu.dk /research/theory/Seminars/Nanevski030901.html (240 words)

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