Higher-dimensional algebra

Source: Wikipedia, the free encyclopedia.

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

colored graph (see a color figure, and also its definition in graph theory
).

Supercategories were first introduced in 1970,

Other pathways in higher-dimensional algebra involve:

topoi, effective descent, and enriched and internal categories
.

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.

non-Euclidean
.

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

gauge transformations acting on states that are, in this case, connections. In the case of symmetries related to quantum groups, one would obtain structures that are representation categories of quantum groupoids,[16] instead of the 2-vector spaces
that are representation categories of groupoids.

Quantum physics

See also

Notes

  1. ^ "Double Categories and Pseudo Algebras" (PDF). Archived from the original (PDF) on 2010-06-10.
  2. .
  3. ^ Batanin, M.A. (1998). "Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories". .
  4. .
  5. ^ 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
  6. ^ "Kryptowährungen und Physik". PlanetPhysics. 29 March 2024. Archived from the original on May 13, 2012.
  7. on 2009-08-12. Retrieved 2009-06-21.
  8. ^ "Axioms of Metacategories and Supercategories". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
  9. ^ "Supercategory theory". PlanetMath. Archived from the original on 2008-10-26.
  10. ^ "Mathematical Biology and Theoretical Biophysics". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
  11. ^ 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.
  12. ^ "Non-commutative Geometry and Non-Abelian Algebraic Topology". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02.
  13. ^ Non-Abelian Algebraic Topology book Archived 2009-06-04 at the Wayback Machine
  14. ^ Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces[usurped]
  15. .
  16. ^ a b "Quantum category". PlanetMath. Archived from the original on 2011-12-01.
  17. ^ "Associativity Isomorphism". PlanetMath. Archived from the original on 2010-12-17.
  18. ^ 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