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Topic: Coproduct category theory

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category theory

abelian categories

limit category theory

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kernel category theory

preadditive categories

product category theory

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Baire category theorem

dual category theory

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	Category of sets - Wikipedia, the free encyclopedia
		In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions.
		Because of Russell's paradox, which shows assuming the existence of the set of all sets leads to a contradiction, the object class of Set is a proper class, and thus the category is large.
		The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.
	en.wikipedia.org /wiki/Set_(category_theory) (315 words)

	Preadditive category - Wikipedia, the free encyclopedia
		In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.
		Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object.
		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.
	en.wikipedia.org /wiki/Preadditive_category (1340 words)

	Encyclopedia :: encyclopedia : Category theory (Site not responding. Last check: 2007-10-30)
		Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
		These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for constructive mathematics.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
	www.hallencyclopedia.com /Category_theory (2343 words)

	PlanetMath: dual category
		For example, a coproduct is a product on the opposite category; this can be seen by looking at the commutative diagram that completely specifies a coproduct, and noting that it is the same as the diagram specifying a product with the arrows reversed.
		A cokernel is a kernel in the opposite category.
		This is version 5 of dual category, born on 2002-02-25, modified 2004-03-29.
	planetmath.org /encyclopedia/DualCategory.html (232 words)

	Category theory
		Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logic s, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
		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.
	www.nebulasearch.com /encyclopedia/article/Category_theory.html (3211 words)

	Category Theory
		Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.
		Category theory is, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures.
		From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics.
	www.science.uva.nl /~seop/archives/spr2006/entries/category-theory (11783 words)

	Category theory - Wikibooks, collection of open-content textbooks
		The notion of category being established as that which gives precision to the concept of domain of mathematical discourse, the formalization of the precise notion corresponding to the intuitive idea of the interrelation or connection between different domains is now considered.
		It is a functor, however, from the category of groups and surjective homomorphisms to the category of groups and all homomorphisms, because a surjective homomorphism does not necessarily map the centre surjectively.
		For example, in the category of sets, the coproduct becomes the disjoint union; in the category of groups it is the free product; and in a pre-ordered set regarded as a category, the coproduct is the least upper bound.
	en.wikibooks.org /wiki/Category_theory (4088 words)

	Addition - Wikipedia, the free encyclopedia
		In set theory, addition is then extended to larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers.
		In category theory, the disjoint union is a kind of coproduct, so coproducts are perhaps the most abstract of all the generalizations of addition.
		Some coproducts are named to evoke their connection with addition; see Direct sum and Wedge sum.
	en.wikipedia.org /wiki/Addition (5534 words)

	Ars Mathematica » Blog Archive » Opinions of Category Theory
		Category theory does not help me at all when I am trying to figure out the long term behaviour of some function given by an ODE - so I don’t use it when I am doing that.
		Your group theory comment is a straw man. category theory is not the natural domain of discourse of the Sylow theorems (or, never underestimate the power of a theorem that counts something).
		While category theory is the natural language of sets with structure, it doesn’t usually capture the “thingness” of any particular class of objects, the quality that makes those objects what they are (I probably sound like Heiddeger now).
	www.arsmathematica.net /archives/2006/06/24/opinions-of-category-theory (3620 words)

	Re: Cobig, Coproduct, and Comma (Site not responding. Last check: 2007-10-30)
		Date: Mon, 20 Mar 89 15:32:11 CST >Cobig, Coproduct, and Comma Vaughan Pratt 3/19/89 >Formally a comma category is most slickly described as a lax pullback.
		I gave a brief calculus of comma categories in: --, The categorical comprehension scheme, Category theory, Homology theory and their Applications III, Lecture Notes in Mathematics 99, Springer-Verlag, New York 1969, 242-312.
		The general theory of the properties of lax limits in 2-categories was discussed independently by Street and me in various publications.
	www.cis.upenn.edu /~bcpierce/types/archives/1989/msg00038.html (334 words)

	MATHS: Category Theory (Site not responding. Last check: 2007-10-30)
		Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic.
		A Category is a mixture of an algebra and a directed graph.
		Category theory shows that in most known algebras and logistic systems, there is a way to construct an equivalent.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	Category:Category theory - Wikipedia, the free encyclopedia
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		There are 13 subcategories to this category shown below (more may be shown on subsequent pages).
		There are 114 pages in this section of this category.
	en.wikipedia.org /wiki/Category:Category_theory (77 words)

	Product and Coproduct
		If the category is concrete, and the index is finite, the product is simply the cartesian product, with the usual component projections.
		In the category of sets, the coproduct is the disjoint union.
		In the category of abelian groups, rings, or r modules, the coproduct of finitely many components is the same as the product.
	www.mathreference.com /cat,prod.html (1148 words)

	Product (category theory) - Free net encyclopedia
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
		Given the Set (the category of sets), the product in the category theoretic sense is the cartesian product.
		A distributive category is one in which this morphism is actually an isomorphism.
	www.netipedia.com /index.php/Product_(category_theory) (632 words)

	SEP: Category Theory
		Category theory thus affords philosophers and logicians much to use and reflect upon.
		Category theory also bears on more general philosophical questions.
		Landry, E. and Marquis, J.-P., 2005, "Categories in Context: Historical, Foundational and philosophical", Philosophia Mathematica, 13, 1–43.
	plato.stanford.edu /entries/category-theory (11786 words)

	week202
		A rig category is basically the most general sort of category in which we can "add" and "multiply" as we do in a ring - but without negatives, hence the missing letter "n".
		He gracefully leads the reader from the very basics of category theory straight to the current battle front of weak n-categories, emphasizing throughout how operads automatically take care of the otherwise mind-numbing thicket of "coherence laws" that inevitably infest the subject.
		It relates the category whose objects are 2-manifolds with a circle as boundary, and whose morphisms are 3-manifolds with corners going between these, to a braided monoidal category "freely generated by a quasitriangular Hopf algebra object".
	math.ucr.edu /home/baez/week202.html (4106 words)

	Category Theory for Computer Science
		Cartesian closed categories and the simplytyped lambda calculus.
		Using Category Theory to Design Implicit Conversions and Generic Operators.
		Category theory in programming language semantics and design
	www.daimi.au.dk /~nygaard/CTfCS (620 words)

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