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Pre-Abelian category - Wikipedia, the free encyclopedia category theory, a functor is called left exact if it preserves all finite limits and right exact if it preserves all finite colimits. Every pre-Abelian category is of course an additive category, and many basic properties of these categories are described under that subject. closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the en.wikipedia.org /wiki/Preabelian_category

PlanetMath: dual category More generally, an inverse limit is a direct limit on the opposite category; for this reason, it is sometimes called a colimit. This is version 5 of dual category, born on 2002-02-25, modified 2004-03-29. 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. www.planetmath.org /encyclopedia/OppositeCategory.html

Cartesian closed category - Wikipedia, the free encyclopedia Substitute categories have therefore been considered: the category of compactly generated Hausdorff spaces is cartesian closed, as is the category of Frölicher spaces. In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed. The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category. en.wikipedia.org /wiki/Cartesian_closed_category

Monoidal category - Wikipedia, the free encyclopedia In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). A monoidal category may be regarded as a bicategory with one object. Mac Lane, Saunders (1997), Categories for the Working Mathematician (2nd ed.). en.wikipedia.org /wiki/Algebra/set_analogy   (560 words)

Category:Mathematics - Wikipedia, the free encyclopedia In mathematics, the Pythagorean theorem or Pythagoras' theorem, is a relation in Euclidean geometry between the three sides of a right angled triangle. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation. If you are interested in learning mathematics, there are several books on the subject at http://www.wikibooks.org, at both the grade school and college level. en.wikipedia.org /wiki/Category:Mathematics   (356 words)

Complete lattice - Wikipedia, the free encyclopedia These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets. Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction. When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one. en.wikipedia.org /wiki/Complete_lattice   (2072 words)

Dual (category theory) - Wikipedia, the free encyclopedia In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms. For example the category of affine schemes is equivalent to the opposite of the category of commutative rings. en.wikipedia.org /wiki/Dual_(category_theory)   (511 words)

Nominal category - Wikipedia, the free encyclopedia A nominal category or a nominal group is a group of objects or ideas that can be collectively grouped on the basis of shared, arbitrary characteristic. This page was last modified 21:40, 5 May 2004. Nominal categories of data are thus most commonly compared to ordinal and ratio data, to see if nominal categories play a role in determining these other factors. en.wikipedia.org /wiki/Nominal_category   (278 words)

Category theory - Wikipedia, the free encyclopedia Such a process is called a functor, and it associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, in connection with algebraic topology. en.wikipedia.org /wiki/Category_theory   (2348 words)

Category 5 cable - Wikipedia, the free encyclopedia Category 5 cable, commonly known as Cat 5, is an unshielded twisted pair type cable designed for high signal integrity. Cat 5 cables are often used in structured cabling for computer networks such as Fast Ethernet, although they are also used to carry many other signals such as basic voice services, token ring, and ATM (at up to 155 Mbit/s, over short distances). Generally solid core cable is used for connecting between the wall socket and the socket in the patch panel whilst stranded cable is used for the patch leads between hub/switch and patch panel socket and between wall port and computer. en.wikipedia.org /wiki/Category_5_cable   (641 words)

Saffir-Simpson Hurricane Scale - Wikipedia, the free encyclopedia Severity categories are scaled somewhat lower than the Saffir-Simpson Scale, with a severity category 2 tropical cyclone being roughly equivalent to a Saffir-Simpson category 1 hurricane. It does not take into account rainfall or location, which means a Category 3 hurricane that hits a major city will likely do far more damage than a Category 5 hurricane that hits a rural area. For other meanings of Category 4, see Category 4 (disambiguation). en.wikipedia.org /wiki/Saffir-Simpson_Hurricane_Scale   (597 words)

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. 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. 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. en.wikipedia.org /wiki/Category_theory   (2348 words)

Simplicial set - Wikipedia, the free encyclopedia Simplicial sets form a category usually denoted s Set or just S whose objects are simplicial sets and whose morphisms are natural transformations between them. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphisms satisfies the axioms for a Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets. en.wikipedia.org /wiki/Simplicial_set   (2348 words)

Additive category - Wikipedia, the free encyclopedia Ab is preadditive because it is a closed monoidal category, and the biproduct in Ab is the finite direct sum. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors, and most interesting functors studied in all of category theory are adjoints. Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject. en.wikipedia.org /wiki/Additive_category   (2348 words)

Dual (category theory) - Wikipedia, the free encyclopedia This is immediately useful, when one can identify the opposite category in concrete terms. This is a special case, since partial orders correspond to a certain kind of category in which Mor( A, B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). en.wikipedia.org /wiki/Categorial_duality   (2348 words)

Preadditive category An additive category is a preadditive category with all biproducts. A preadditive category is a category that is enriched over the monoidal category of abelian groups. If C and D are categories and D is preadditive then the functor category Fun(C D) is also preadditive because natural transformations can be added in a natural If C is preadditive too then the category C D) of additive functors and all natural between them is also preadditive. www.freeglossary.com /Module_category   (1324 words)

Ryle: Mind/Body Problem is a Category Mistake A category mistake is a mistake about the kind of thing that some thing essentially is. www.ukans.edu /~acudd/phil140-s8/tsld008.htm   (1324 words)

Encyclopedia: Category:Mathematics Mathematics and architecture – Mathematics and education – Mathematics of musical scales Fields in applied mathematics use knowledge of mathematics to solve real world problems. Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. www.nationmaster.com /encyclopedia/Category:Mathematics   (1504 words)

Encyclopedia: Grammatical number In linguistics, number is a grammatical category that specifies the quantity of a noun or affects the form of a verb or other part of speech depending on the quantity of the noun to which it refers. Grammatical number is distinct from the use of numerals to specify the exact quantify of a noun; number is usually vague. English is typical of languages that have singular and plural number. www.nationmaster.com /encyclopedia/Grammatical-number   (1504 words)

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