Higher-dimensional algebra

Higher-dimensional algebra

"This article is about" higher-dimensional algebra and supercategories, "respectively, in generalized category theory, and supercategory theory or meta-mathematics".

In higher-dimensional algebra [cite journal|last = Batanin|first = MA|title = Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories|publisher=Royal Society|journal=Advances in Mathematics|volume = 136|issue = 1|date = 1998-06-01|pages = 39–103|doi = 10.1006/aima.1998.1724] , a "double groupoid" is a generalisation of a 'one-dimensional' groupoid to two dimensions.Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds. Similarly, a supercategory is a higher-dimensional concept that generalises the notion of category--regarded as any structure which is an interpretation of Lawvere's ETAC axioms; it can be alternatively regarded as a natural extension of a meta-category, or as a multi-graph.

Double groupoids were first introduced by Ronald Brown [http://en.wikipedia.org/wiki/Ronald_Brown_(mathematician)] in 1976, and were further developed towards applications in non-Abelian Algebraic Topology [ [http://www.bangor.ac.uk/~mas010/nonab-a-t.html "Non-Abelian Algebraic Topology" book] ] .

Supercategories were first introduced in 1970 as explained in ref. [ [http://planetmath.org/encyclopedia/Supercategories3.html Supercategory theory] ] , and were subsequently developed for applications in Theoretical Physics (especially Quantum Field Theory) and Mathematical Biophysics.

References

5. Ronald Brown, Higgins, P. J. and R. Sivera. 2007-2008, vol. 1 [http://www.bangor.ac.uk/~mas010/nonab-a-t.html "Non-Abelian Algebraic Topology" book] , (vol. 2 "in preparation"); [http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf downloadable PDF:]

* [http://en.wikipedia.org/wiki/Ronald_Brown_(mathematician) Ronald Brown] and C.B. Spencer, Double groupoids and crossed modules, "Cahiers Top. G'eom.Diff.", 17 (1976) 343-362.

*Ronald Brown and G.H. Mosa, Double categories, thin structures and connections, "Theory and Applications of Categories", 5 (1999), 163-175.

*Ronald Brown. 2002. "Categorical Structures for Descent and Galois Theory". Fields Institute, September 23-28, 2002.

*Ronald Brown, From groups to groupoids: a brief survey, Bull. LMS, 19 (1987) 113-134, gives some of the history of groupoids, namely the origins in work of Brandt on quadratic forms, and an indication of later work up to 1987, with 160 references. These have been updated slightly in the downloadable version, available as ref. [http://www.bangor.ac.uk/r.brown/groupoidsurvey.pdf] ]

* [http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory] is a web article with lots of references explaining how the groupoid concept has to led to notions of higher dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology.

* Ronald Brown and P.J. Higgins, On the algebra of cubes, "J. Pure & Applied Algebra", 21 (1981), 233--260.

* [http://www.shef.ac.uk/~pm1kchm/gt.html General theory of Lie groupoids and Lie algebroids, K.C.H. Mackenzie, CUP, 2005]

* [http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids, Ronald Brown, Booksurge 2006] revised and extended edition of a book previously published in 1968 and 1988. e-version available.

* [http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090 Galois theories, F. Borceux, G. Janelidze, CUP, 2001] , shows how generalisations of Galois theory lead to Galois groupoids.

*I. C. Baianu, 1970. Organismic Supercategories: II. On Multistable Systems, "Bulletin of Mathematical Biophysics"., 32,539-561.

*I. C. Baianu and M. Marinescu, 1974. On A Functorial Construction of (M,R)- Systems., "Revue Roumaine de Mathe'matiques Pures et Applique'es", 19: 388-391.

*I. C. Baianu, 1986--1987. Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), "Mathematical Models in Medicine", vol. 7., Pergamon Press, New York, 1513-1577; CERN "Preprint" No. EXT-2004-072.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Higher category theory — is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Contents 1 Strict higher categories 2 Weak higher… …   Wikipedia

  • algebra, modern — ▪ mathematics Introduction also called  abstract algebra        branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers (real number), complex numbers (complex number), matrices (matrix), and… …   Universalium

  • algebra — /al jeuh breuh/, n. 1. the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. 2. any of… …   Universalium

  • Algebra — This article is about the branch of mathematics. For other uses, see Algebra (disambiguation). Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from… …   Wikipedia

  • algebra, elementary — Introduction       branch of mathematics that deals with the general properties of numbers and the relations between them. Algebra is fundamental not only to all further mathematics and statistics but to the natural sciences, computer science,… …   Universalium

  • Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

  • Geometric algebra — In mathematical physics, a geometric algebra is a multilinear algebra described technically as a Clifford algebra over a real vector space equipped with a non degenerate quadratic form. Informally, a geometric algebra is a Clifford algebra that… …   Wikipedia

  • Comparison of vector algebra and geometric algebra — Vector algebra and geometric algebra are alternative approaches to providing additional algebraic structures on vector spaces, with geometric interpretations, particularly vector fields in multivariable calculus and applications in mathematical… …   Wikipedia

  • Seven-dimensional cross product — In mathematics, the seven dimensional cross product is a binary operation on vectors in a seven dimensional Euclidean space. It is a generalization of the ordinary three dimensional cross product. The seven dimensional cross product has the same… …   Wikipedia

  • Nichols algebra — The Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”