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Topic: Counit

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	CoUnit
		CoUnit is not meant for testing components written in Java; we have JUnit for that.
		CoUnit helps you to test the bits in between: All the code that lives in the pipelines, transforming XML (SAX events).
		CoUnit is based on XSLTUnit, which is a single stylesheet developed by Eric van der Vlist of Dyomeda.
	new.cocoondev.org /main/117/104.html (1325 words)

	[No title]
		A A, with counit, which is a map of regular bimodules.
		A2 This latter diagram is commutative because the square on the left expresses the module property of ffi, and the square on the right express the counit property of ffl.
		Because the composition of maps across the top and down the right of this diagram is simply the definition of 5m5, and the composition of maps down the left and across the bottom is just 5 (by the unit property of 1A), we see that 5m5 5.
	www.math.purdue.edu /research/atopology/Abrams/abrams-cotensor.txt (3329 words)

	290300 (Site not responding. Last check: 2007-11-03)
		This problem can be solved by the simple algebraic process of adding a unit or counit.
		We will investigate a class of algebraic structures that include algebras with unit and coalgebras with counit as special cases.
		From a categorical point of view these structures appear as very natural generalizations of the notion of algebra and coalgebra as it is know from classical algebra.We will also develope a theory of quantization for these algebraic structures.
	www.math.uit.no /seminar/ABSTR/290300.htm (116 words)

	Coalgebra (Site not responding. Last check: 2007-11-03)
		Accordingly, the map is called the comultiplication of C and is the counit of C. Examples
		Delta(s)s otimess quad mbox{ and } quad epsilon(s) 1 quad mbox{ for all }s in S. By linearity, both and can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication and counit (you may want to check this to get used to the axioms).
		A subspace D of C is called a subcoalgebra if (D)DD, in that case, D is itself a coalgebra, with the restriction of to D as counit.
	read-and-go.hopto.org /Abstract-algebra/Coalgebra.html (866 words)

	Theory CoUnit (Isabelle2005: October 2005)
		} theory CoUnit* imports Main begin text {* See discussion in: L C Paulson.
		} consts counit* :: i codatatype "counit" = Con ("x ∈ counit") inductive_cases ConE: "Con(x) ∈ counit" -- {* USELESS because folding on @{term "Con(xa) == xa"} fails.
		} by (auto elim!: counit.free_elims) lemma counit_eq_univ: "counit* = quniv(0)" -- {* Should be a singleton, not everything!
	isabelle.in.tum.de /library/ZF/ex/CoUnit.html (456 words)

	Citebase - Hopf C*-algebras
		In this paper we introduce the notion of a Hopf C-algebra and construct the counit* and antipode.
		The leading example is of course the C*-algebra of continuous, vanishing at infinity functions on a locally compact group.
		We include several formulas for the counit and antipode which are familiar from Hopf algebra theory.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9907030 (1174 words)

	Hopf Algebras
		Throughout this section, A is a bialgebra with comultiplication and counit
		Recall lemma 4.16 that explains the nature of counit map on the level of tensor products of modules.
		Corollary 4.23 does a similar thing for antipodes.
	www.maths.warwick.ac.uk /~rumynin/rings2002/ln/node37.html (120 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		I am interested in the following situation: a contravariant functor adjoint to its own dual, with the unit and counit being the same morphism, but _not_ an iso.
		The canonical example is the contravariant internal hom on a cartesian (or just symmetric monoidal) closed category, [(_) -> A] for some object A. My question is: is this typical, or are there (interesting) examples of such adjunctions that do not come from exponentials?
		At 01:29 PM 2/4/97 -0400, you wrote: >I am interested in the following situation: a contravariant functor >adjoint to its own dual, with the unit and counit being the same >morphism, but _not_ an iso.
	www.mta.ca /~cat-dist/catlist/1999/adj-dual (239 words)

	[No title]
		(We are using the notations and definitions of [2].) We are assuming that the unit and counit of A"hare obtained by simply extend* ing those of A, so that there will be no difference in the notations for these maps, whet *her considered on A or on A"h.
		Then U is a Hopf algebra over Z with respect to th* e diagonal defined by Xn (Xn) = Xi Xn-i: i=0 The counit* ffl is the map taking a polynomial in the noncommuting variables Xit* *o its constant term.
		The values of the antipode S for U may be inductively computed using the * *formulas nX Xn S(Xi)Xn-i= XiS(Xn-i) = 0; i=0 i=0 which hold for all n > 0.
	www.math.purdue.edu /research/atopology/Duflot/quantumgrp2.txt (3750 words)

	Math Forum Discussions
		>definition with the units and counits (I will not state it here)..
		Alas, he promises that the names of unit and counit will be easier to
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?messageID=3676120&tstart=0 (621 words)

	Preprint No. 334 (Site not responding. Last check: 2007-11-03)
		Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (Delta,epsilon).
		We do not require Delta(1) = 1 o 1 nor multiplicativity of the counit epsilon.
		Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that Rep A becomes a monoidal category with unit object given by the cyclic A-submodule E := (A --> epsilon) c A^ (A^ denoting the dual weak bialgebra).
	www-sfb288.math.tu-berlin.de /abstractNew/334 (127 words)

	Citebase - Axioms for Weak Bialgebras
		Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (Δ,eps).
		Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that RepA becomes a monoidal category with unit object given by the cyclic A-submodule E := (A --> eps) subset hat A (hat A denoting the dual weak bialgebra).
		Under these monoidality axioms E and E := (eps <-- A) become commuting unital subalgebras of hat A which are trivial if and only if the counit eps is multiplicative.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9805104 (1283 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		@unit@ and @counit@ Given functors @f@ and @g@, @Adjunction f g@ implies @Monad (g `'O'` f)@ and @'Comonad' (f `'O'` g)@.
		deComp instance (Adjunction f g) => Comonad (O f g) where extract = counit.
		deComp instance Adjunction ((,) a) ((->) a) where unit t = \x -> (x,t) counit (x,f) = f x
	www.eyrie.org /~zednenem/2004/hsce/Control/Functor/Adjunction.hs (119 words)

	LicensedPreludeExts - The Haskell Wiki (Site not responding. Last check: 2007-11-03)
		unit = leftAdjunct id counit = rightAdjunct id leftAdjunct f = fmap f.
		unit t = \x -> (x,t) counit (x,f) = f x
		Some examples of use: swing map :: forall a b.
	www.haskell.org /hawiki/LicensedPreludeExts (1267 words)

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