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# Topic: Category theory

 Category Theory (Stanford Encyclopedia of Philosophy) 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. plato.stanford.edu /entries/category-theory   (11810 words)

 PlanetMath: category theory Category theory gives us tools for analyzing such functors: we can talk about natural transformations of functors, and in fact we can use these to assemble the category of functors from one category to another into a category, provided certain set-theoretic constraints are met (universes are a tool used to address these set-theoretic difficulties). The fundamental theorem of Galois theory is that the functor from a subgroup of the Galois group of a field to its fixed field is an equivalence of categories. Since categories are able to deal with isomorphisms and functors are compatible with the category structure, functors are generally a good way of attacking the problem “are these two objects isomorphic?”;. planetmath.org /encyclopedia/CategoryTheory.html   (1640 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. 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. 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. www.math.niu.edu /~rusin/known-math/index/18-XX.html   (286 words)

 Learn more about Category theory in the online encyclopedia.   (Site not responding. Last check: 2007-10-02) 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. One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings. www.onlineencyclopedia.org /c/ca/category_theory.html   (2963 words)

 Kids.net.au - Encyclopedia Category theory - Category theory is half-jokingly known as "abstract nonsense". Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories. One may check that the map from the category of Hausdorff topological spaces with a distinguished point to the category of groups is functorial: a topological (homo/iso)morphism will naturally correspond to a group (homo/iso)morphism. www.kids.net.au /encyclopedia-wiki/ca/Category_theory   (2107 words)

 Category Theory Category theory is a general mathematical theory of structures and sytems of structures. 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". Given these simple facts, it remains to be seen whether category theory should be "on the same plane", so to speak, with set theory, whether it should be considered seriously as providing a foundational alternative to set theory or whether it is foundational in a different sense altogether. www.seop.leeds.ac.uk /archives/spr2001/entries/category-theory   (3068 words)

 Structuralism, Category Theory and Philosophy of Mathematics   (Site not responding. Last check: 2007-10-02) 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. 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 is essentially anti-platonistic, for it undermines the received idea that the meaning of any mathematical concept is fixed by referring it to the context of a unique absolute universe of sets. cs.wwc.edu /~aabyan/CII/strctcat.htm   (8116 words)

 Math The basis for the theory is laid by first showing that the object-to-object "heteromorphisms" between the objects of different categories (e.g., insertion of generators as a set to group map) can be rigorously treated within category theory. B is the concrete universal for the property of being a subset of A and a subset of B. I argue that category theory is relevant to foundations as the theory of concrete universals. Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal. www.ellerman.org /Davids-Stuff/Maths/Math.htm   (2368 words)

 alpheccar's blog - Index Category Theory and Haskell 3 : Algebras and Monads In my previous post, I explained that with category theory you can define some concepts in such a way that they can be used in several different contexts. "Category theory" is an expression that is generally frightening people. www.alpheccar.org   (954 words)

 Good Math, Bad Math This is one of the last posts in my series on category theory; and it's a two parter. Category theory provides a good framework for defining linear logic - and for building a Curry-Howard style type system for describing computations with state that evolves over time. For me, the frustrating thing about learning category theory was that it seemed to be full of definitions, but that I couldn't see why I should care. scienceblogs.com /goodmath/goodmath/category_theory   (1459 words)

 Categories Home Page 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. 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 at the Isle of Thorns was held from July 7 to 12, 1996. www.mta.ca /~cat-dist   (3848 words)

 Category Theory Category theory studies structural aspects of mathematics that are common to many fields of mathematics: e.g., algebra, topology, functional analysis, logic, and computer science. Category theory is a mathematical language which arose in the study of limits for universal coefficient theorems in Cech cohomology by Eilenberg and Mac Lane (1942); so the topic has its origins in some sophisticated topology. The category version of the definition of a group: A group is a category with one object in which all the morphisms are isomorphisms. education.wichita.edu /alagic/nextpage/categories.htm   (1383 words)

 Category theory and consciousness   (Site not responding. Last check: 2007-10-02) Category theory has been proposed as a new approach to the deep problems of modern physics, in particular quantization of General Relativity. Category theory might provide the desired systematic approach to fuse together the bundles of general ideas related to the construction of quantum TGD proper. Category theory might also have natural applications in the general theory of consciousness and the theory of cognitive representations. www.physics.helsinki.fi /~matpitka/articles/acategory.html   (482 words)

 The Reference Frame: Category theory and physics String theory is first of all a physical theory, and it should be studied because of physical motivations - the primary physical motivation is to locate the right ideas and equations that describe the real world. 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. 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. motls.blogspot.com /2004/11/category-theory-and-physics.html   (3827 words)

 Stephen's Software Blog category theory   (Site not responding. Last check: 2007-10-02) Category theory is a branch of abstract algebra which generalizes both set theory and graph theory. Because category theory is a generalization of graph theory, it is natural to ask if it would be useful for working on SCMs, because the version graph is central to understanding SCMs. Saunders points out that slice categories (he calls them "comma categories") were long an "secret tool in the arsenal of knowledgeable experts." This motivates me to look at slice categories in the category of versions and patches. turnbull.sk.tsukuba.ac.jp /Blogs/Software/CategoryTheory   (375 words)

 Category Theory Course 2004   (Site not responding. Last check: 2007-10-02) Examples are the category of rings with ring homomorphisms, and the category of topological spaces with continous maps as morphisms. Category theory also provides a formal setting for reasoning about interactions between categories using the concepts of functors and natural transformations. However, even though applications of category theory to computer science and logic are covered in the course, a background in computer science is not necessary. www.itu.dk /people/mogel/catcourse   (454 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 be read by researchers and students in computing science. Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another. www.case.edu /artsci/math/wells/pub/ctcs.html   (1715 words)

 Category Theory Research of Victor Porton - pure category theory + theory of formulas That is I have reduced category theory to theory of dependencies. Category of elements of a category - given a category to whose morphisms correspond binary relations, build a category of elements of this category; Category of elements of endomorphisms - continuing the previous article defined elements of endomorphisms of a (pre)category, also defined subelements, loopless elements, etc. (It is a generalization of graph theory.); www.mathematics21.org /category-theory.html   (465 words)

 Topos Theory at Chicago | The n-Category Café These current topics lie at the intersection of algebraic geometry, topology, logic, and higher category theory, and may be of interest to a wider audience. Here are two lectures on topos theory, part of the University of Chicago’s Fall 2006 Category Theory Seminar, and serving as a warmup for ∞-topoi. Topos theory, the unification of this diversity of viewpoints, is a beautiful field of mathematics, but one which it can be difficult to get a handle on. golem.ph.utexas.edu /category/2006/10/topos_theory_at_chicago.html   (703 words)

 Category Theory Science, Directory   (Site not responding. Last check: 2007-10-02) Computational Category Theory Project The aim of this 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. www.wacofdn.org /d2RjXzI2OTM1.aspx   (287 words)

 Category Theory and Homotopy Theory   (Site not responding. Last check: 2007-10-02) Category theory was introduced in 1947 to give a richer language than that of set theory, which would be better able to express the structures of homotopy and homology theory then being revealed in the work of Cartan, Eilenberg, Mac Lane, Whitehead and others. This language and theory was soon found to have great usefulness in other branches of pure mathematics such as algebra, algebraic geometry, logic and more recently in computer science. The basic areas of research in category theory at Bangor are directed towards achieving a greater understanding of the categorical structure and interrelationships between the various objects studied by algebraic topology and homological algebra. www.informatics.bangor.ac.uk /public/mathematics/research/cathom/cathom1.html   (214 words)

 Category Theory for Computing Science   (Site not responding. Last check: 2007-10-02) 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. Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another. www.cwru.edu /artsci/math/wells/pub/ctcs.html   (1730 words)

 Computational Category Theory | FreeTechBooks.com Someone with a computing background who wishes to learn category theory should have recourse to standard texts, some of which are listed later, but could well find this book a helpful companion text. This book should be helpful to computer scientists wishing to understand the computational significance of theorems in category theory and the constructions carried out in their proofs. It should also be of interest to mathematicians familiar with category theory - they may not be aware of the computational significance of the constructions arising in categorical proofs. www.freetechbooks.com /about469.html   (519 words)

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