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Topic: Coequalizers

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	Coequalizer (Site not responding. Last check: 2007-10-25)
		In mathematics a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category.
		The coequalizer is a special kind of colimit in category theory.
		In the category of sets the coequalizer of two functions f g : X → Y is the quotient of Y by the equivalence relation generated by the relations f (x) = g (x) for all x in X.
	www.freeglossary.com /Coequalizer (580 words)

	Cokernel - Wikipedia, the free encyclopedia
		The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0
		Like all coequalizers, the cokernel q : Y → Q is necessarily an epimorphism.
		In such a category, the coequalizer of two morphisms f and g (if it exists) is just the cokernel of their difference:
	en.wikipedia.org /wiki/Cokernel (522 words)

	PlanetMath: equalizer
		Reversing all the arrows in the previous paragraphs, we have the dual notion of an equalizer: that of a coequalizer.
		A coequalizer is an epimorphism (and conversely, an epimorphism that is also a coequalizer is called a regular epimorphism).
		Kernels and cokernels are necessarily unique by the universality of equalizers and coequalizers.
	planetmath.org /encyclopedia/Equalizer.html (287 words)

	Epimorphism - Wikipedia, the free encyclopedia
		Every coequalizer is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers.
		The converse, namely that every epimorphism be a coequalizer, is not true in all categories.
		An extremal epimorphism is an epimorphism that has no monomorphism as a second factor, unless that monomorphism is an isomorphism.
	en.wikipedia.org /wiki/Epimorphism (1548 words)

	[No title]
		Formally, C B R is a coequalizer of a right action of B on C and the given acti* *on of B on R. Remark 4.8.
		In the symmetric case, we may take our j given coequalizer diagrams to be the same and compose the j-fold power, regarded as a functor to the category of j-objects in C, with the orbit functor.
		The latte* r is constructed as a coequalizer* in C and is a left adjoint, so preserves coequaliz* *ers.
	www.math.purdue.edu /research/atopology/Elmendorf-Kriz-Mandell-May/ekmm.txt (20587 words)

	AMCA: Poly-bicategories by Jurgen Koslowski (Site not responding. Last check: 2007-10-25)
		This is indeed the case, provided the domain poly-bicategory is representable and closed, in the sense that every 1-cell has both a left and a right adjoint.
		Previously, only the case of the domain being terminal and the codomain being representable and locally having reflexive equalizers and reflexive coequalizers was well-understood.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/e/q/42.htm (319 words)

	[No title]
		The unit object of pro* -C is the constant pro-object with value the unit object in C. We only consider te nsor structures on pro-C that are levelwise tensor structures inherited from a tensor structure on C. If C is a cocomplete category, then pro-C is a cocomplete category [16, 11.1] *.
		We recall the description of arbitrary direct sums and of coequalizers in pro-C. Let A be an indexing set and let Xff2 pro-C for ff 2 A be a set of pro-object* *s in C.`Let Iffbe the cofiltered indexing category of the pro-object Xff.
		The coequalizer is the pro-object {coeq(Xa ' Ya)} obtained by forming the coequalizer levelwise in C. We now consider how direct sums and tensor`products interact.
	hopf.math.purdue.edu /Fausk-Isaksen/t-model.txt (10164 words)

	On Sifted Colimits And Generalized Varieties - Ad, Rosick (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Filtered colimits, i.e., colimits over schemes D such that D-colimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that D-colimits in Set commute with finite products.
		An important example: reflexive coequalizers are sifted colimits.
		Generalized varieties are defined as free completions of small categories under sifted-colimits (analogously to finitely accessible categories which are free filtered-colimit...
	citeseer.ist.psu.edu /418095.html (415 words)

	McLarty: Elementary Categories, Elementary Toposes (Site not responding. Last check: 2007-10-25)
		We have the dual notions of cocones and colimits and the special cases of coproducts, coequalizers and pushouts.
		For a CCC C, the functor -xA:C->C (i.e., "product with A") preserves initial objects, coproducts, coequalizers and pushouts [it preserves all colimits since it is left adjoint to -^A].
		This (by a basic categorical argument using nothing special about toposes) implies that any coequalizer of is a coequalizer of .
	gtps.math.cmu.edu /cebrown/notes/mclarty.html (7588 words)

	[No title]
		X(n) and the commutative diagram y y is a pushout X#n.
		An equalizer (coequalizer) of an f 2 Mor (X; Y) and 0XY is said t* *o be a kernel_ (cokernel_) of f.
		F * is a coequalizer* of (d1; d0) = (Fm; aeT) and the preceding observations dualize.] 1-1 x1.
	hopf.math.purdue.edu /WarnerG/warner-book.txt (13171 words)

	Theses from Umeå University: 245 - Categorical Unification (Site not responding. Last check: 2007-10-25)
		This thesis deals with different aspects towards many-valued unification which have been studied in the scope of category theory.
		The main motivation of this investigation comes from the fact that in logic programming, classical unification has been identified as the provision of coequalizers in Kleisli categories of term monads.
		Continuing in that direction, we have used categorical instrumentations to generalise the classical concept of a term.
	www.diva-portal.org /umu/theses/abstract.xsql?dbid=245 (402 words)

	Bar Constructions
		X_{1} --> X_{0} --> X_{-1}, it follows from acyclicity that this portion of the simplicial structure is a split coequalizer, so that the augmentation map is the usual augmentation to its set of path components.
		However, in practice many coequalizers of the type shown above split, and we can refer to triple products XYZ without essential ambiguity if the coequalizers ending with XY and YZ split.
		If M is an operad, then the free M-bimodule monad F is given by the assignment X --> M.X.M, and if X is itself a bimodule, we obtain a bar construction B(F, F, X).
	math.ucr.edu /home/baez/universal/bar.html (2334 words)

	Abstract: Abstracting Refinements for Transformation (Site not responding. Last check: 2007-10-25)
		This paper introduces a semantics for rewriting that is independent of the data being rewritten and which, nevertheless, models key concepts such as substitution which are central to rewriting algorithms.
		We demonstrate the naturalness of this construction by showing how it mirrors the usual treatment of algebraic theories as coequalizers of monads.
		We also demonstrate its naturalness by showing how it captures several canonical forms of rewriting.
	www.informatik.uni-bremen.de /~cxl/papers/abstracts/njc04b.html (84 words)

	On Induced Representations of Lie Algebras, Groups and Coalgebras
		Let (H, G, F, L, i, R, e) be a double adjoint situation, and let G be a category with epimorphic images, in which epimorphisms are coequalizers.
		is a commutative diagram in a category in which epimorphisms are coequalizers, and that α is epi, β is monic.
		Thus, by the universal property of coequalizers, there is a unique morphism ε: B →i>C such that εα = γ.
	www.cse.unsw.edu.au /~billw/mathresearch/induced.html (3274 words)

	KAN
		Kan will achieve this by using a specification language based upon category theory whose algebraic nature fits well with the problems to be solved.
		At first sight, it would seem natural to model quotients as coequalizers but this is too restrictive as all objects would be forced to belong to the same category.
		For example, cosets are quotients of a group by a subgroup but form a set and not a group.
	www.cs.le.ac.uk /people/ah83/kan (543 words)

	Semantics Lunch
		Only basic category theory (functors, natural transformations, limits) will be assumed.
		Adjunctions whose counits are coequalizers and presentations of finitary enriched monads.
		Abstract: HOPLA (Winskel et al) is a Higher-Order Process LAnguage which, by design, has a clean and natural denotational semantics.
	www.cl.cam.ac.uk /users/pes20/semanticslunch.html (1294 words)

	[No title]
		coequalizer[0] == -2)) { viewbox+="1 is a coequalizer.\n1 coequalizer found."; } else if (objectsButton.isSelected()) { for (int i=0; i
		coequalizerChoice.getSelectedItem().equals("Check all objects")) { // call checkCoequalizer method coeq.checkEqualizer(i, alpha, beta); viewbox+=coeq.getOutput(); } } if (coeq.getNumEqualizers() == 0) viewbox+="No coequalizers found.\n"; else if (coeq.getNumEqualizers() == 1) viewbox+="1 coequalizer found.\n"; else viewbox+=Integer.toString(coeq.getNumEqualizers()) + " coequalizers found.\n"; } else { coeq.setSupressOutput(false); coeq.equalizer(coequalizer, alpha, beta); viewbox+=coeq.getOutput(); } if (coeq.getEndoPassed()) viewbox+="WARNING: Endomorphism limit reached!
		Results may be inaccurate!\n"; JOptionPane.showInternalMessageDialog(this, viewbox, "Coequalizers", JOptionPane.INFORMATION_MESSAGE); } private void closeButton_actionPerformed(ActionEvent e) { try {this.setClosed(true);} catch (PropertyVetoException PVE) {} } private void pathsButton_action() { coequalizerPathLabel.setEnabled(true); coequalizerPath.setEnabled(true); coequalizerObjectLabel.setEnabled(false); coequalizerChoice.setEnabled(false); } private void objectsButton_action() { coequalizerPathLabel.setEnabled(false); coequalizerPath.setEnabled(false); coequalizerObjectLabel.setEnabled(true); coequalizerChoice.setEnabled(true); } }
	mathcs.mta.ca /research/rosebrugh/gdct/java/CoequalizerFrame.java (128 words)

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