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Topic: Category:Category theory

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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. We take "theorem" here to mean provable from the axioms of the elementary theory of an abstract category. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms. en.wikipedia.org /wiki/Dual_(category_theory)   (511 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)

Structuralism, Category Theory and Philosophy of Mathematics Category theory is the language best suited for this type of representation because it avoids the incommensurability problems which result from the Tarskian semantics essential to mathematical logic and model- theory for which satisfaction relations and truth definitions can only be defined for a specific language and the structure used to explicate the semantics. Category theory furnishes such a formulation through the concept of topos, and its formal counterpart, local set theory.[Bell,238] Any topos may be regarded as a mathematical domain of discourse or 'world' in which mathematical concepts can be interpreted and mathematical constructions performed. Category theory, as we shall see later, is compatible with this position since it takes as fundamental arrows or morphisms, which are generalizations of functions. www.mmsysgrp.com /strctcat.htm   (7237 words)

Abstract algebra:Category theory - Wikibooks Category theory is the study of categories, which are collections of objects and morphisms (or arrows), or from one object to another. The category whose objects are smooth (differentiable,topological) manifolds, and morphisms are smooth (differentiable,continuous) maps. is a category with the same objects, and all the arrows reversed. en.wikibooks.org /wiki/Abstract_algebra:Category_theory   (309 words)

Category Theory (Reading Course) Category theory is a kind of network algebra which provides a general framework for describing mathematical objects and their interrelations. The standard example of a category is the family Ens of sets and functions with composition defined as ordinary composition of functions and the identity arrow is the identity function. A category is a non-null family of arrows with a binary law of composition which is only partially defined; that is, not every pair of arrows is composable. www.georgetown.edu /faculty/kainen/category.html   (1030 words)

Luboš Motl's reference frame: Category theory and physics Category theory often resembles linguistics (or even postmodern literary criticism): it is a science about arrows between different objects and about creating new objects from these arrows, but it does not really care too much whether the objects exist and what are their real properties. Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor, many people have claimed that everything that happens in a classical theory has a counterpart in the "corresponding" quantum theory. The mathematically oriented part of the string theory community is mostly excited by category theory. motls.blogspot.com /2004/11/category-theory-and-physics.html   (3454 words)

Category Theory For it is in his thesis that Lawvere proposed the idea of developing the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of theories, that is the logical aspects of mathematics. Category theory reveals that many of these constructions are in fact special cases of objects in a category with what is called a "universal property". On the one hand, it is certainly the task of philosophy to clarify the general epistemological status of category theory and, in particular, its foundational status. plato.stanford.edu /entries/category-theory   (7029 words)

18: Category theory, homological algebra Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. A full, wide-ranging text on category theory is by Borceux, Francis: "Handbook of categorical algebra", 3 vol (1: Basic category theory; 2: Categories and structures; 3: Categories of sheaves) (Encyclopedia of Mathematics and its Applications, 50-2.) Cambridge University Press, Cambridge, 1994. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology. www.math.niu.edu /~rusin/known-math/index/18-XX.html   (286 words)

Limit (category theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. Typical examples of categories that are not complete are categories with some "size restriction": the category of finite groups or the category of finite-dimensional vector spaces over a fixed field. Consequently, the category J is usually a small category and has fewer elements than the category C. www.worldhistory.com /wiki/L/Limit-(category-theory).htm   (1880 words)

Category Theory for Computing Science Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. This book is a textbook in basic category theory, written specifically to beread by researchers and students in computing science.   A topos is a kind of generalized set theory in which the logic is intuitionistic instead of classical. www.cwru.edu /artsci/math/wells/pub/ctcs.html   (1730 words)

Categories Home Page Category Theory at the Isle of Thorns was held from July 7 to 12, 1996. Category Theory Symposium The symposium was a special session of the Canadian Mathematical Society Summer 1998 Meeting June 13-15, 1998 at University of New Brunswick (Saint John) Saint John, New Brunswick, Canada and was part of a larger programme including plenary talks, one of which was given by S. Schanuel (SUNY Buffalo). Category Theory 2000 The international summer conference in category theory was held at Villa Olmo, Como, Italy from Sunday 16th July to Saturday 22nd July 2000. www.mta.ca /~cat-dist   (3476 words)

Amazon.com: Basic Category Theory for Computer Scientists (Foundations of Computing): Books: Benjamin C. Pierce Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. My interest is in general category theory, and I bought this because I have a BS in CS and thought I'd find plenty of familiar examples. If you would like to know the first step of Category Theory and you are in CS realm, this book is the one you have to try. www.amazon.com /exec/obidos/tg/detail/-/0262660717?v=glance   (1060 words)

CATEGORY THEORY AT MCGILL Category Theory at McGill The category theorists that constitute our group are, in order of their joining the Department, Jim Lambek, Marta Bunge, Michael Barr and Michael Makkai,with the addition of Robert Seely and Thomas Fox as Adjunct Professors. Category Theory Category theorists are conceptual mathematicians of a special kind. Current Research Areas in Category Theory at McGill Three areas deserve attention because of the novelties they bring and because they are part of a truly international joint effort. www.math.mcgill.ca /bunge/ctatmcgill.html   (549 words)

Category Theory Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations. This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines. www.andrew.cmu.edu /course/80-413-713   (149 words)

Citations: Basic category theory - Poigne (ResearchIndex) The definitive text on category theory is MacLane s book [88] 5.1 Graphs A graph consists of a set of objects O (vertices) a set of arrows A (edges) and a pair of functions# 0, # 1 : A , called the source and target functions respectively. Other notions from category theory can be defined similarly in an O enriched setting, but we shall not need it here. Proposition 5.2 An O category C induce the functor hom(I; Gamma) C O. If C moreover is equipped with a monoidal structure is an O functor, then a monoidal structure on the functor hom(I; Gamma) is induced by the map n 1 : 1 hom(I; I).... citeseer.ist.psu.edu /context/322244/0   (853 words)

Open Directory - Science: Math: Algebra: Category Theory The Computational Category Theory Project- The aim of the project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures. Category Theory - This expository article is an entry in the Stanford Encyclopedia of Philosophy. CT Category Theory - Section of the e-print arXiv dealing with category theory, including such topics as: enriched categories, topoi, abelian categories, monoidal categories, homological algebra. dmoz.org /Science/Math/Algebra/Category_Theory   (326 words)

[Inquiry] Re: Category Theory A 'category' is a graph with two additional functions: In treating a category C, we usually A category (as distinguished from a metacategory) will stderr.org /pipermail/inquiry/2003-July/000623.html   (298 words)

Kernel (category theory) - Wikipedia, the free encyclopedia In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field). en.wikipedia.org /wiki/Kernel_(category_theory)   (298 words)

Kernel (mathematics) - Wikipedia, the free encyclopedia In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f. Kernels in abstract algebra are general constructions which measure the failure of a homomorphism or function to be injective. The kernel pair of a morphism f is defined as a pullback of f with itself. en.wikipedia.org /wiki/Kernel_(mathematics)   (298 words)

Image (category theory) - Wikipedia, the free encyclopedia In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism f can be expressed as follows: In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets. The image of f is often denoted by im f. en.wikipedia.org /wiki/Image_(category_theory)   (298 words)

Coproduct In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Coproducts are actually special cases of colimit s in category theory. The coproduct can be defined as the colimit of a discrete subcategory in C. www.worldhistory.com /wiki/c/coproduct.htm   (298 words)

Normal morphism - Wikipedia, the free encyclopedia In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup. A normal category is a category in which morphisms are normal, whenever reasonable. en.wikipedia.org /wiki/Binormal   (292 words)

Category of topological spaces - Wikipedia, the free encyclopedia The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. This is a category because the composition of two continuous maps is again continuous. to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. en.wikipedia.org /wiki/Top_(category_theory)   (292 words)

Elementary Category Theory Lambda the Ultimate For those interested in a brief introduction to category theory, James Cheney has recently posted some PDF slides titled Category Theory for Dummies on his home page. Recently I started learning CT, and I'm trying to express the property "set A is contained within set B" (in the Set Category) using CT language only, but I'm quite stuck. About the closest you can get in pure CT and using a generic notion of Set Theory (and, possibly intuitionism and Choice are also relevant), is that a subset is iso to a `subobject', which is an equivalence class of monos. lambda-the-ultimate.org /node/view/39   (1262 words)

Centre de Recherche en Théorie des Catégories -- Montréal Category Theory for Computing Science (Ordering information for another classic text by M Barr and C Wells, the current version published by the CRM.) Category Theory - an expository article from the Stanford Encyclopedia of Philosophy. Category Theory at McGill A personal account by Marta Bunge. www.math.mcgill.ca /triples   (453 words)

Introduction to Category Theory This text differs from most other introductions to category theory in the calculational style of the proofs (especially in Chapter~\ref{ch:constructions} and Appendix~\ref{moreonadjointness}), the restriction to applications within algorithmics, and the omission of many additional concepts and facts that I consider not helpful in a first introduction to category theory. In these notes we present the important notions from category theory. A Gentle Introduction to Category Theory - the calculational approach wwwhome.cs.utwente.nl /~fokkinga/mmf92b.html   (181 words)

Categories F(X) Category theory is popular among algebraic topologists. A functor F from a category C to a category D is a map from the set of objects of C to the set of objects of D together with a map from the set Hom(X,Y) for any objects X,Y of C to Hom(F(X),F(Y)). A representation of a group, if we think of a group as a category as Sibley suggests, is just a functor from that category to the category Vect of vector spaces. math.ucr.edu /home/baez/categories.html   (2546 words)

Structure (category theory) - Wikipedia, the free encyclopedia In category theory structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structures. The analogue in category theory is the Yoneda lemma. Starting with a given class of algebraic structure, such as groups, one can build the category in which the objects are groups and the morphisms are group homomorphisms: that is, of structures on one type, and mappings respecting that structure. en.wikipedia.org /wiki/Structure_(category_theory)   (397 words)

Pushout (category theory) - Wikipedia, the free encyclopedia In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout is the categorical dual of the pullback. The pushout of f and g is the union of X and Y together with the inclusion morphisms from X and Y. en.wikipedia.org /wiki/Pushout_(category_theory)   (894 words)

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