Study of categorified structures
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
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 .
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 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 .
Nonabelian algebraic topology
See
Nonabelian algebraic topology
Applications
Theoretical physics
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 SUq (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.
Quantum physics
See also
Notes
^
"Double Categories and Pseudo Algebras" (PDF) . Archived from
the original (PDF) on 2010-06-10.
^ Brown, R.; Loday, J.-L. (1987). "Homotopical excision, and Hurewicz theorems, for n -cubes of spaces".
Proceedings of the London Mathematical Society . 54 (1): 176–192.
CiteSeerX
10.1.1.168.1325 .
doi :
10.1112/plms/s3-54.1.176 .
^
Batanin, M.A. (1998).
"Monoidal Globular Categories As a Natural Environment for the Theory of Weak n -Categories" .
Advances in Mathematics . 136 (1): 39–103.
doi :
10.1006/aima.1998.1724 .
^ Lawvere, F. W. (1964).
"An Elementary Theory of the Category of Sets" . Proceedings of the National Academy of Sciences of the United States of America . 52 (6): 1506–1511.
Bibcode :
1964PNAS...52.1506L .
doi :
10.1073/pnas.52.6.1506 .
PMC
300477 .
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.
^ Lawvere, F. W. (1969b).
"Adjointness in Foundations" . Dialectica . 23 (3–4): 281–295.
CiteSeerX
10.1.1.386.6900 .
doi :
10.1111/j.1746-8361.1969.tb01194.x . Archived from
the original on 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
^ Brown, Ronald; Higgins, Philip; Sivera, Rafael (2011).
Nonabelian Algebraic Topology .
arXiv :
math/0407275 .
doi :
10.4171/083 .
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.
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 . .