Higher-dimensional algebra
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Higher-dimensional categories
A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category.[1][2][3]
A higher level concept is thus defined as a
Supercategories were first introduced in 1970,
Other pathways in higher-dimensional algebra involve:
Double groupoids
In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions,[11] and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.
Double groupoids were first introduced by Ronald Brown in Double groupoids and crossed modules (1976),[11] and were further developed towards applications in nonabelian algebraic topology.[12][13][14][15] A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.
Nonabelian algebraic topology
See Nonabelian algebraic topology
Applications
Theoretical physics
In
Quantum physics
- Quantum Algebraic Topology
- Quantum geometry
- Quantum gravity
- Quantum group
- Topological quantum field theory
- Local quantum field theory
See also
- Timeline of category theory and related mathematics
- Higher category theory
- Ronald Brown
- Lie algebroid
- Double groupoid
- Anabelian geometry
- Noncommutative geometry
- Categorical algebra
- Grothendieck's Galois theory
- Grothendieck topology
- Topological dynamics
- Categorical dynamics
- Crossed module
- Pseudoalgebra
- 2-ring
- Lie n-algebra
Notes
- ^ "Double Categories and Pseudo Algebras" (PDF). Archived from the original (PDF) on 2010-06-10.
- .
- ^ Batanin, M.A. (1998). "Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories". .
- PMID 16591243.
- ^ Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in Proc. Conf. Categorical Algebra – La Jolla., Eilenberg, S. et al., eds. Springer-Verlag: Berlin, Heidelberg and New York., pp. 1–20. http://myyn.org/m/article/william-francis-lawvere/ Archived 2009-08-12 at the Wayback Machine
- ^ "Kryptowährungen und Physik". PlanetPhysics. 29 March 2024. Archived from the original on May 13, 2012.
- doi:10.1111/j.1746-8361.1969.tb01194.x. Archived from the originalon 2009-08-12. Retrieved 2009-06-21.
- ^ "Axioms of Metacategories and Supercategories". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
- ^ "Supercategory theory". PlanetMath. Archived from the original on 2008-10-26.
- ^ "Mathematical Biology and Theoretical Biophysics". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
- ^ a b c Brown, Ronald; Spencer, Christopher B. (1976). "Double groupoids and crossed modules". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 17 (4): 343–362.
- ^ "Non-commutative Geometry and Non-Abelian Algebraic Topology". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
- ^ Non-Abelian Algebraic Topology book Archived 2009-06-04 at the Wayback Machine
- ^ Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces[usurped]
- ISBN 978-3-03719-083-8.
- ^ a b "Quantum category". PlanetMath. Archived from the original on 2011-12-01.
- ^ "Associativity Isomorphism". PlanetMath. Archived from the original on 2010-12-17.
- ^ a b c Morton, Jeffrey (March 18, 2009). "A Note on Quantum Groupoids". C*-algebras, deformation theory, groupoids, noncommutative geometry, quantization. Theoretical Atlas.
Further reading
- Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Vol. Tracts Vol 15. European Mathematical Society. )
- Brown, R.; Mosa, G.H. (1999). "Double categories, thin structures and connections". Theory and Applications of Categories. 5: 163–175. CiteSeerX 10.1.1.438.8991.
- Brown, R. (2002). Categorical Structures for Descent and Galois Theory. Fields Institute.
- Brown, R. (1987). "From groups to groupoids: a brief survey" (PDF). hdl:10338.dmlcz/140413. This give some of the history of groupoids, namely the origins in work of Heinrich Brandton quadratic forms, and an indication of later work up to 1987, with 160 references.
- Brown, Ronald (2018). "Higher Dimensional Group Theory". groupoids.org.uk. Bangor University. A web article with many references explaining how the groupoid concept has 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". .
- Mackenzie, K.C.H. (2005). General theory of Lie groupoids and Lie algebroids. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. ISBN 978-0-521-49928-6. Archived from the originalon 2005-03-10.
- Brown, R. (2006). Topology and Groupoids. ISBN 978-1-4196-2722-4. Revised and extended edition of a book previously published in 1968 and 1988. E-version available from website.
- Borceux, F.; Janelidze, G. (2001). Galois theories. Cambridge University Press. OCLC 1167627177. Archived from the original on 2012-12-23. Shows how generalisations of Galois theorylead to Galois groupoids.
- Baez, J.; Dolan, J. (1998). "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes". S2CID 18857286.
- Baianu, I.C. (1970). "Organismic Supercategories: II. On Multistable Systems" (PDF). The Bulletin of Mathematical Biophysics. 32 (4): 539–61. PMID 4327361.
- 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 (ed.). Mathematical Models in Medicine. Vol. 7. .
- "Higher dimensional Homotopy". PlanetPhysics. Archived from the original on 2009-08-13.
- Janelidze, George (1990). "Pure Galois theory in categories". Journal of Algebra. 132 (2): 270–286. .
- Janelidze, George (1993). "Galois theory in variable categories". Applied Categorical Structures. 1: 103–110. S2CID 22258886..