Higher-dimensional algebra

This article is about higher-dimensional algebra and supercategories in generalized category theory, super-category theory, and also its extensions in nonabelian algebraic topology and metamathematics.

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

Double groupoids, fundamental groupoids, 2-categories, categorical QFTs and TQFTs

In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, 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). 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. 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. Other pathways in HDA involve: bicategories, homomorphisms of bicategories, variable categories (aka, indexed, or parametrized categories), topoi, effective descent, enriched and internal categories, as well as quantum categories and quantum double groupoids. In the latter case, by considering fundamental groupoids defined via a 2-functor 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"; 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, instead of the 2-vector spaces that are representation categories of groupoids.

Double categories, Category of categories and Supercategories

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

Double groupoids were first introduced by Ronald Brown in 1976, in ref. and were further developed towards applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.

Nonabelian algebraic topology

Many of the higher dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theory, and also of Nonabelian Algebraic Topology (NAAT) which generalises to higher dimensions ideas coming from the fundamental group. Such algebraic structures in dimensions greater than 1 develop the nonabelian character of the fundamental group, and they are in a precise sense ‘more nonabelian than the groups' . These noncommutative, or more specifically, nonabelian structures reflect more accurately the geometrical complications of higher dimensions than the known homology and homotopy groups commonly encountered in classical algebraic topology. An important part of nonabelian algebraic topology is concerned with the properties and applications of homotopy groupoids and filtered spaces. Noncommutative double groupoids and double algebroids are only the first examples of such higher dimensional structures that are nonabelian. The new methods of Nonabelian Algebraic Topology (NAAT) ``can be applied to determine homotopy invariants of spaces, and homotopy classification of maps, in cases which include some classical results, and allow results not available by classical methods". Cubical omega-groupoids, higher homotopy groupoids, crossed modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy of filtered spaces, higher dimensional space structures, the construction of the fundamental groupoid of a topos E in the general theory of topoi, and also in their physical applications in nonabelian quantum theories, and recent developments in quantum gravity, as well as categorical and topological dynamics. Further examples of such applications include the generalisations of noncommutative geometry formalizations of the noncommutative standard models via fundamental double groupoids and spacetime structures even more general than topoi or the lower-dimensional noncommutative spacetimes encountered in several topological quantum field theories and noncommutative geometry theories of quantum gravity.

A fundamental result in NAAT is the generalised, higher homotopy van Kampen theorem proven by R. Brown which states that ``the homotopy type of a topological space can be computed by a suitable colimit or homotopy colimit over homotopy types of its pieces''. A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include statements for 2-categories and a topos of topoi [1]. Important results in HDA are also the extensions of the Galois theory in categories and variable categories, or indexed/`parametrized' categories. The Joyal-Tierney representation theorem for topoi is also a generalisation of the Galois theory. Thus, indexing by bicategories in the sense of Benabou one also includes here the Joyal-Tierney theory.

See also

• Timeline of category theory and related mathematics
• Higher category theory
• Category theory
• Ronald Brown
• Algebraic topology
• Lie algebroid
• Groupoid
• 2-category
• Double groupoid
• Quantum Algebraic Topology
• Seifert–van Kampen theorem
• Chain complex
• Homological algebra
• Homology theory
• Cohomology
• Galois theory
• Anabelian geometry
• Quantum geometry
• Noncommutative geometry
• Abstract algebra
• Categorical algebra
• Grothendieck's Galois theory
• Grothendieck topology
• Topological dynamics
• Topological quantum field theory
• Local quantum field theory
• Categorical dynamics
• Quantum group
• Quantum gravity
• Categorical group
• Fundamental group
• Crossed module
• Pursuing stacks
• Esquisse d'un Programme
• Metatheory
• Metalogic
• Metamathematics
• Colored graphs
• Multicategory
• Enriched category

Further reading

• (Downloadable PDF available)
• 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.
• . 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.
• Revised and extended edition of a book previously published in 1968 and 1988. E-version available from http://www.kagi.com
• Shows how generalisations of Galois theory lead to Galois groupoids.
• George Janelidze, Pure Galois theory in categories, J. Alg. 132:270–286, 1990.
• Galois theory in variable categories., by George Janelidze, Dietmar Schumacher and Ross Street, in APPLIED CATEGORICAL STRUCTURESVolume 1, Number 1, 103--110, doi:10.1007/BF00872989.de:Gruppoid (Kategorientheorie)