Higher-dimensional algebra

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This article is about higher-dimensional algebra and supercategories in generalized category theory, super-category theory, and also its extensions in metamathematics^[1]. Supercategories were first introduced in 1970,^[2] and were subsequently developed for applications in Theoretical Physics (especially Quantum Field Theory and Topological quantum field theory) and Mathematical Biology or Mathematical Biophysics.^[3] In higher-dimensional algebra, a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions^[4], and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.

Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds)^[5]. In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean. A first step towards defining higher dimensional algebras is the concept of 2-category, followed by the more `geometric' concept of double category^[6][7].

A higher level concept is that of a category of categories, or super-category which generalises to higher dimensions the notion of category – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC)^{[8][9][10][11]}. Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category,^[12] multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).

Double groupoids were first introduced by Ronald Brown in 1976, in ref.^[13] and were further developed towards applications in nonabelian algebraic topology^{[14][15][16][17]}.

[edit] See also

• Higher category theory
• Category theory
• Algebraic topology
• Seifert–van Kampen theorem
• Abstract algebra
• Categorical algebra
• Esquisse d'un Programme
• Grothendieck's Galois theory
• Metatheory
• Metalogic
• Metamathematics
• Colored graphs
• Multicategory
• Enriched category

[edit] Notes

[edit] Further reading

• Brown, R.; Higgins, P.J.; Sivera, R. (2008). Non-Abelian Algebraic Topology. 1. http://www.bangor.ac.uk/~mas010/nonab-a-t.html.  (Downloadable PDF)
• Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules,". Cahiers Top. Geom. Diff. 17: 343–362. 
• Brown, R.; Mosa, G.H. (1999). "Double categories, thin structures and connections". Theory and Applications of Categories 5: 163–175. 
• Brown, R. (2002). Categorical Structures for Descent and Galois Theory. Fields Institute. 
• Brown, R. (1987). "From groups to groupoids: a brief survey". Bulletin of the London Mathematical Society 19: 113–134. doi:10.1112/blms/19.2.113. http://www.bangor.ac.uk/r.brown/groupoidsurvey.pdf.  This give some of the history of groupoids, namely the origins in work of Heinrich Brandt on quadratic forms, and an indication of later work up to 1987, with 160 references.
• Brown, R.. "Higher dimensional group theory". http://www.bangor.ac.uk/r.brown/hdaweb2.htm. . 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.
• Brown, R.; Higgins, P.J. (1981). "On the algebra of cubes". Journal of Pure and Applied Algebra 21: 233–260. doi:10.1016/0022-4049(81)90018-9. 
• Mackenzie, K.C.H. (2005). General theory of Lie groupoids and Lie algebroids. Cambridge University Press. http://www.shef.ac.uk/~pm1kchm/gt.html. 
• R., Brown (2006). Topology and groupoids. Booksurge. http://www.bangor.ac.uk/r.brown/topgpds.html.  Revised and extended edition of a book previously published in 1968 and 1988. E-version available at PlanetPhysics.org and Bangor.ac.uk
• Borceux, F.; Janelidze, G. (2001). Galois theories. Cambridge University Press. http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090.  Shows how generalisations of Galois theory lead to Galois groupoids.
• Baez, J.; Dolan, J. (1998). "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes". Advances in Mathematics 135: 145–206. doi:10.1006/aima.1997.1695. 
• Baianu, I.C. (1970). "Organismic Supercategories: II. On Multistable Systems". Bulletin of Mathematical Biophysics 32: 539–561. doi:10.1007/BF02476770. 
• Baianu, I.C.; Marinescu, M. (1974). "On A Functorial Construction of (M, R)-Systems". Revue Roumaine de Mathématiques Pures et Appliquées 19: 388–391. 
• Baianu, I.C. (1987). "Computer Models and Automata Theory in Biology and Medicine". in M. Witten. Mathematical Models in Medicine. 7. Pergamon Press. 1513–1577. CERN Preprint No. EXT-2004-072.. http://cogprints.org/3687/. 
• "Higher dimensional Homotopy @ PlanetPhysics". http://planetphysics.org/encyclopedia/HigherDimensionalHomotopy.html.