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.


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