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] }

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).^{ [3] }^{ [4] } Ll.

,^{ [5] }^{ [6] } Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category,^{ [7] } 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,^{ [8] } and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.^{ [9] }

Other pathways in higher-dimensional algebra involve: bicategories, homomorphisms of bicategories, variable categories (*aka*, 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,^{ [10] } 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 1976, in ref.^{ [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.

In quantum field theory, there exist quantum categories.^{ [16] }^{ [17] }^{ [18] } and quantum double groupoids.^{ [19] } 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 *";^{ [19] } 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.

- 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
- Areas of application in quantum physics:

- ↑ 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. Archived from the original on 2009-08-12. Retrieved 2009-06-21. - ↑ 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".
- ↑ 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. - ↑ "Kryptowährungen und Physik – Planetphysics". Archived from the original on 2009-08-14. Retrieved 2009-03-02.
- ↑ Supercategory theory @ PlanetMath
- ↑ "Kryptowährungen und Physik – Planetphysics". Archived from the original on 2009-08-14. Retrieved 2009-03-02.
- ↑ Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules".
*Cahiers Top. Géom. Diff*.**17**: 343–362. - 1 2 Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules" (PDF).
*Cahiers Top. Géom. Diff*.**17**: 343–362. Archived from the original (PDF) on 2008-07-24. - ↑ "Kryptowährungen und Physik – 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, R.; et al. (2009).
*Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces*(in press).^{[ permanent dead link ]} - 1 2 http://planetmath.org/encyclopedia/QuantumCategory.html Quantum Categories of Quantum Groupoids
- ↑ http://planetmath.org/encyclopedia/AssociativityIsomorphism.html Rigid Monoidal Categories
- ↑ "A Note on Quantum Groupoids". 2009-03-18.
- 1 2 http://theoreticalatlas.wordpress.com/2009/03/18/a-note-on-quantum-groupoids/ March 18, 2009. A Note on Quantum Groupoids, posted by Jeffrey Morton under C*-algebras, deformation theory, groupoids, noncommutative geometry, quantization

- Brown, R.; Higgins, P.J.; Sivera, R. (2011).
*Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids*. Tracts Vol 15. European Mathematical Society. arXiv: math/0407275 . doi:10.4171/083. ISBN 978-3-03719-083-8. (Downloadable PDF available) - Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules".
*Cahiers Top. Géom. 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" (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, R. "Higher dimensional group theory".. 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*. Cambridge University Press. Archived from the original on 2005-03-10. - R., Brown (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. 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. - Baianu, I.C. (1970). "Organismic Supercategories: II. On Multistable Systems" (PDF).
*Bulletin of Mathematical Biophysics*.**32**(4): 539–61. doi:10.1007/BF02476770. PMID 4327361.External link in`|journal=`

(help) - 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*.**7**. Pergamon Press. pp. 1513–1577. CERN*Preprint*No. EXT-2004-072. ASIN 0080346928 ASIN 0080346928. - "Higher dimensional Homotopy @ PlanetPhysics". Archived from the original on 2009-08-13.
- George Janelidze, Pure Galois theory in categories, J. Alg. 132:270–286, 1990.
- Janelidze, George (1993). "Galois theory in variable categories".
*Applied Categorical Structures*.**1**: 103–110. doi:10.1007/BF00872989..

**Category theory** formalizes mathematical structure and its concepts in terms of a labeled directed graph called a *category*, whose nodes are called *objects*, and whose labelled directed edges are called *arrows*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

A **CW complex** is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The *C* stands for "closure-finite", and the *W* for "weak" topology. A CW complex can be defined inductively.

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of *objects* and *arrows*, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

In mathematics, the **Seifert–Van Kampen theorem** of algebraic topology, sometimes just called **Van Kampen's theorem**, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

In mathematics, the **Eckmann–Hilton argument** is an argument about two unital magma structures on a set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting magma demonstrated to be commutative monoid. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in a 1962 paper.

In mathematics, and especially in homotopy theory, a **crossed module** consists of groups *G* and *H*, where *G* acts on *H* by automorphisms (which we will write on the left, , and a homomorphism of groups

In mathematics, **higher category theory** is the part of category theory at a *higher order*, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.

This is a **timeline of category theory and related mathematics**. Its scope is taken as:

In mathematics, a **2-group**, or **2-dimensional higher group**, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of *n*-groups. In some of the literature, 2-groups are also called **gr-categories** or **groupal groupoids**.

In mathematics, **directed algebraic topology** is a refinement of algebraic topology for **directed spaces**, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in Concurrency, Network traffic control, General Relativity, Noncommutative Geometry, Rewriting Theory, and Biological systems.

In mathematics, especially in higher-dimensional algebra and homotopy theory, a **double groupoid** generalises the notion of groupoid and of category to a higher dimension.

In mathematics, **R-algebroids** are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups..

**Ronald Brown** is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.

In mathematics, in the theory of Hopf algebras, a **Hopf algebroid** is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf *k*-algebroids. If *k* is a field, a commutative *k*-algebroid is a cogroupoid object in the category of *k*-algebras; the category of such is hence dual to the category of groupoid *k*-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry. They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra *H* is a Hopf algebroid over its separable subalgebra *H ^{L}*. The antipode axioms have been changed by G. Böhm and K. Szlachányi in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.

In mathematics, a **quantum groupoid** is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations, in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories, Hopf algebroids etc.

In algebraic topology, the **fundamental groupoid** is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.

[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen Theorem it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,]

In mathematics, **nonabelian algebraic topology** studies an aspect of algebraic topology that involves higher-dimensional algebras.

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