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Topic: Coequaliser

	Pre-Abelian category
		exists (this is the kernel), as does the coequaliser (this is the cokernel).
		Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre-Abelian category actually has all equalisers and coequalisers.
		; similarly, their coequaliser is the cokernel of their difference.
	www.teachtime.com /en/wikipedia/p/pr/pre_abelian_category.html (836 words)

	Preadditive category - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-21)
		That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism.
		There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets.
		Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers.
	encyclopedia.worldsearch.com /preadditive_category.htm (1303 words)

	Practical Foundations of Mathematics
		and w is the coequaliser of the parallel pair.
		Recall that general coequalisers are computed in several steps (Lemma 5.6.11), not all of which are needed this case, but we still need to consider stable directed unions in order to form the transitive closure.
		The coequaliser is also peculiar to this situation in another respect, namely that it only works for unary recursion, cf the special properties of unary algebra in Section 3.8.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s64.html (2520 words)

	Pre-Abelian category - Preagelian_category - Dangeruss-Industries.com
		given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is the kernel), as does the coequaliser (this is the cokernel).
		Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an Abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.
		We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
	www.dangeruss-industries.com /Preabelian_category_x_Preagelian_category_x.html (421 words)

	Practical Foundations of Mathematics
		Q is the coequaliser of some pair W \rightrightarrows A then it is also the coequaliser of its kernel pair.
		The coequaliser q:A\twoheadrightarrow Q of a congruence K\rightrightarrows A is known as its quotient and written Q = A/K.
		Colimits by duality Although this way of constructing coequalisers is not in general available for infinitary operations, most of those of interest (meets, joins, limits and colimits) are defined by universal properties.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s56.html (1603 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-10-21)
		Since most of the interesting phenomena may be observed more clearly in the concrete cases of pullback, equaliser, pushout and coequaliser, we postpone the abstract definition to Section 7.3.
		This chapter is an account of first order logic, originally motivated by the needs of homological algebra - the understanding of sets, functions and relations came later.
		The extension to while in Section 6.4 shows that general coequalisers in Set are much more complicated than equalisers or quotients.
	www.geocities.com /yury_bendersky/b/f/s50.html (340 words)

	Practical Foundations of Mathematics
		The most general unifier is not the coequaliser amongst all algebras.
		We shall discuss coequalisers amongst arbitrary algebras in Section 7.4, and characterise the subcategory of free algebras in Proposition 7.5.3(a).
		Each step of the execution has replaced one unification problem by another which is equivalent to it, the last being an assignment of terms to independent variables, which has a coequaliser.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s65.html (1156 words)

	[No title]
		X is a strong colimit of the diagram if W x X is the colimit of {W x Xi} for each W 2 C. We define strong coproducts and strong coequalisers as special cases of this, in the obvious way.
		XX reflects strong coequalisers, it is enough to show that t* he above diagram becomes a strong coequaliser* after applying f*.
		Xf is an equivalence, with inverse given by Q. Moreover, the coequaliser in XY that defines QU is a strong coequaliser.
	hopf.math.purdue.edu /Strickland/st-fsfg.txt (9406 words)

	Colimit
		The general definition of a colimit is given on the limit page.
		Special cases of colimits, each dual to a special case of limits, include coproduct[?], coequaliser[?], pushout[?].
		Take this note to be able to access this article instantly from any page
	www.fastload.org /co/Colimit.html (106 words)

	[No title]
		The bit of the (categorical) diagram which does the work is a coequaliser, and Barry Jay observed independently that this says in categorical language that the WHILE program is the "universal loop invariant".
		The categorical structure corresponding to first order logic is a (Heyting) pretopos, which need not have coequalisers of arbitrary parallel pairs.
		Finally, since my more recent work on categorical recursion, I no longer think that a coequaliser diagram is the best way of presenting a WHILE program categorically (though the diagram I would use now expresses the same logical content).
	www.mta.ca /~cat-dist/catlist/1999/cat-while-floyd-hoare (1160 words)

	Practical Foundations of Mathematics
		be a category that has kernel pairs and their coequalisers.
		Y, let q:X\twoheadrightarrow Q be the coequaliser of the kernel pair K\rightrightarrows X of f; by Lemma 5.6.6(b) this is also the kernel pair of q.
		We must show that Q\hookrightarrow Y, so form its kernel pair L\rightrightarrows Q and let P be the coequaliser.
	www.geocities.com /yury_bendersky/b/f/s57.html (1337 words)

	[No title]
		X for each * U 2 U. Theorem 4.1 (van Kampen theorem) If the interiors of the sets of U cover X, then in the above æ-sequence of the cover, c is the coequaliser* of a, b in the category* * of double groupoids with connections.
		In this case the * *coequaliser reduces to a pushout.
		The proof of the theorem is a direct verification of the universal property* * for the coequaliser.
	hopf.math.purdue.edu /BrownR-Kamps-Porter/vkt7.txt (4566 words)

	Tractatus categorico_toposophicus
		h = (q,r) is universal iff r = u;q = ;q and q is the coequaliser (Definition 5.1.1(a) and Proposition 5.6.8ff).
		We shall treat general limits and colimits fully in Section 7.3.
		Free algebras Besides coequalisers, the new feature which arises when we move from posets to categories is the free algebra (Chapter VI).
	toposophicus.blogspot.com (4101 words)

	[No title]
		Subject: categories: co- What are the origins of the co- prefix, as in coproduct, coequaliser..., and who established their use?
		Has anybody ever thought through and written down any guidelines on which of a pair of dual concepts is co-?
		This is also the origin of words such as coordinate (1641), coefficient (ca.
	www.mta.ca /~cat-dist/catlist/1999/co-prefix (5563 words)

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