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.

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 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).^[4][5][6][7] Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category,^[8] multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).

Supercategories were first introduced in 1970,^[9] and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.^[10]

Other pathways in higher-dimensional algebra involve: bicategories, homomorphisms of bicategories, variable categories (also known as indexed or parametrized categories), topoi, effective descent, and enriched and internal categories.

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 are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds).^[11] 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.

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.

See Nonabelian algebraic topology

In quantum field theory, there exist quantum categories.^[16][17][18] and quantum double groupoids.^[18] One can consider quantum double groupoids to be fundamental groupoids defined via a 2-functor, which allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the bicategory Span(Groupoids), and then constructing 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms. At the next step, one obtains cobordisms with corners via natural transformations of such 2-functors. A claim was then made that, with the gauge group SU(2), "the extended TQFT, or ETQFT, gives a theory equivalent to the Ponzano–Regge model of quantum gravity";^[18] similarly, the Turaev–Viro model would be then obtained with representations of SU_q(2). Therefore, one can describe the state space of a gauge theory – or many kinds of quantum field theories (QFTs) and local quantum physics, in terms of the transformation groupoids given by symmetries, as for example in the case of a gauge theory, by the 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.

• 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. arXiv:math/0407275. doi:10.4171/083. ISBN 978-3-03719-083-8. (Downloadable PDF available)
• 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). Bulletin of the London Mathematical Society. 19 (2): 113–134. CiteSeerX 10.1.1.363.1859. doi:10.1112/blms/19.2.113. hdl:10338.dmlcz/140413. 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, 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". Journal of Pure and Applied Algebra. 21 (3): 233–260. doi:10.1016/0022-4049(81)90018-9.
• 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 original on 2005-03-10.
• Brown, R. (2006). Topology and Groupoids. Booksurge. 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. ISBN 978-0-521-07041-6. OCLC 1167627177. Archived from the original on 2012-12-23. 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 (2): 145–206. arXiv:q-alg/9702014. Bibcode:1997q.alg.....2014B. doi:10.1006/aima.1997.1695. S2CID 18857286.
• Baianu, I.C. (1970). "Organismic Supercategories: II. On Multistable Systems" (PDF). The Bulletin of Mathematical Biophysics. 32 (4): 539–61. doi:10.1007/BF02476770. 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. Pergamon Press. pp. 1513–77. ISBN 978-0-08-034692-2. OCLC 939260427. CERN Preprint No. EXT-2004-072. ASIN 0080346928 ASIN 0080346928.
• "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. doi:10.1016/0021-8693(90)90130-G.
• Janelidze, George (1993). "Galois theory in variable categories". Applied Categorical Structures. 1: 103–110. doi:10.1007/BF00872989. S2CID 22258886..