Higher-dimensional algebra
Encyclopedia
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
.
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 morphism
s.
Double groupoid
s 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 group
oids 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 space
s and 2-linear maps for manifolds and cobordism
s. At the next step, one obtains cobordism
s with corners via natural transformation
s 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 representation
s of SU_q(2). Therefore, according to the construction proposed by Jeffrey Morton, 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 group
s, one would obtain structures that are representation categories of quantum groupoids, instead of the 2-vector space
s that are representation categories of groupoids.
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
.
, 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 group
oids and filtered spaces. Noncommutative double groupoid
s and double algebroid
s 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 group
oids, crossed module
s, crossed complexes
and Galois group
oids are key concepts in developing applications related to homotopy of filtered spaces, higher dimensional space structures, the construction of the fundamental group
oid 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 model
s 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 http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz.
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.
on quadratic forms, and an indication of later work up to 1987, with 160 references.. A web article with lots of references explaining how the groupoid concept has to 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 at PlanetPhysics.org and Bangor.ac.uk Shows how generalisations of Galois theory
lead to Galois groupoids.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
(especially quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
) and mathematical biology
Mathematical biology
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology...
or mathematical biophysics
Mathematical biology
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology...
.
Double groupoids, fundamental groupoids, 2-categories, categorical QFTs and TQFTs
In higher-dimensional algebra (HDA), a double groupoidDouble groupoid
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.- Definition :...
is a generalisation of a one-dimensional groupoid
Groupoid
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:...
to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s.
Double groupoid
Double groupoid
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.- Definition :...
s 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
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
of higher category theory
Higher category theory
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.- Strict higher categories :...
, 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 group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
oids defined via a 2-functor allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the bicategory
Bicategory
In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B...
Span(Groupoids), and then constructing 2-Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s and 2-linear maps for manifolds and cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
s. At the next step, one obtains cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
s with corners via natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s 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
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
"; similarly, the Turaev-Viro model would be then obtained with representation
Representation (mathematics)
In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...
s of SU_q(2). Therefore, according to the construction proposed by Jeffrey Morton, one can describe the state space
State space
In the theory of discrete dynamical systems, a state space is a directed graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ = b where the function f defines the dynamical system.State spaces are...
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 group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
s, one would obtain structures that are representation categories of quantum groupoids, instead of the 2-vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s that are representation categories of groupoids.
Double categories, Category of categories and Supercategories
A higher level concept is thus defined as a categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of categories, or super-category, which generalises to higher dimensions the notion of category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
– 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
Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
, can be regarded as natural extensions of the concepts of meta-category
Multicategory
In mathematics , a multicategory is a generalization of the concept of category that allows morphisms of multiple arity...
, multicategory
Multicategory
In mathematics , a multicategory is a generalization of the concept of category that allows morphisms of multiple arity...
, and multi-graph, k-partite graph
Turán graph
The Turán graph T is a graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have subsets of size \lceil n/r\rceil, and r- subsets of size \lfloor n/r\rfloor...
, or colored graph (see a color figure, and also its definition in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
).
Double groupoids were first introduced by Ronald Brown
Ronald Brown (mathematician)
Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:...
in 1976, in ref. and were further developed towards applications in nonabelian algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
. A related, 'dual' concept is that of a double algebroid
Lie algebroid
In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
, and the more general concept of R-algebroid
R-algebroid
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...
.
Nonabelian algebraic topology
Many of the higher dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theoryCategory theory
Category theory is an 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...
, and also of Nonabelian Algebraic Topology (NAAT) which generalises to higher dimensions ideas coming from the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
. 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
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
.
An important part of nonabelian algebraic topology is concerned with the properties and applications of homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
oids and filtered spaces. Noncommutative double groupoid
Double groupoid
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.- Definition :...
s and double algebroid
Lie algebroid
In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
s 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 group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
oids, crossed module
Crossed module
In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H , and a homomorphism of groups...
s, crossed complexes
Timeline of category theory and related mathematics
This is a timeline of category theory and related mathematics. Its scope is taken as:* Categories of abstract algebraic structures including representation theory and universal algebra;* Homological algebra;* Homotopical algebra;...
and Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
oids are key concepts in developing applications related to homotopy of filtered spaces, higher dimensional space structures, the construction of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
oid 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
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
, as well as categorical and topological dynamics
Topological dynamics
In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.- Scope :...
. Further examples of such applications include the generalisations of noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
formalizations of the noncommutative standard model
Noncommutative standard model
In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on the...
s via fundamental double groupoids and spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
structures even more general than topoi
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
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
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
and a topos of topoi http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz.
Important results in HDA are also the extensions of the Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group 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 mathematicsTimeline of category theory and related mathematicsThis is a timeline of category theory and related mathematics. Its scope is taken as:* Categories of abstract algebraic structures including representation theory and universal algebra;* Homological algebra;* Homotopical algebra;...
- Higher category theoryHigher category theoryHigher 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.- Strict higher categories :...
- Category theoryCategory theoryCategory theory is an 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...
- Ronald BrownRonald Brown (mathematician)Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:...
- Algebraic topologyAlgebraic topologyAlgebraic topology is a branch of mathematics which 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.Although algebraic topology...
- Lie algebroidLie algebroidIn mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
- GroupoidGroupoidIn 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:...
- 2-category2-categoryIn category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
- Double groupoidDouble groupoidIn mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.- Definition :...
- Quantum Algebraic TopologyTopological orderIn physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
- Seifert–van Kampen theoremSeifert–van Kampen theoremIn 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 X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
- Chain complexChain complexIn mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
- Homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
- Homology theoryHomology theoryIn mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
- CohomologyCohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
- Galois theoryGalois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
- Anabelian geometryAnabelian geometryAnabelian geometry is a proposed theory in mathematics, describing the way the algebraic fundamental group G of an algebraic variety V, or some related geometric object, determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian...
- Quantum geometryQuantum geometryIn theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales...
- Noncommutative geometryNoncommutative geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
- Abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
- Categorical algebraCategorical algebraIn category theory, a field of mathematics, a categorical algebra is an associative algebra, defined for any locally finite category and commutative ring with unity.It generalizes the notions of group algebra and incidence algebra,...
- Grothendieck's Galois theoryGrothendieck's Galois theoryIn mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry...
- Grothendieck topologyGrothendieck topologyIn category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
- Topological dynamicsTopological dynamicsIn mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.- Scope :...
- Topological quantum field theoryTopological quantum field theoryA topological quantum field theory is a quantum field theory which computes topological invariants....
- Local quantum field theoryLocal quantum field theoryThe Haag-Kastler axiomatic framework for quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local quantum physics of C*-algebra theory. It is therefore also known as Algebraic Quantum Field Theory...
- Categorical dynamicsSymbolic dynamicsIn mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...
- Quantum groupss
- Quantum gravityQuantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
- Categorical group2-groupIn 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...
- Fundamental groupFundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
- Crossed moduleCrossed moduleIn mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H , and a homomorphism of groups...
- Pursuing stacksAlexander GrothendieckAlexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
- Esquisse d'un ProgrammeEsquisse d'un Programme"Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck...
- MetatheoryMetatheoryA metatheory or meta-theory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called metatheorems....
- MetalogicMetalogicMetalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves...
- MetamathematicsMetamathematicsMetamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
- Colored graphGraph coloringIn graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the...
s - MulticategoryMulticategoryIn mathematics , a multicategory is a generalization of the concept of category that allows morphisms of multiple arity...
- Enriched categoryEnriched categoryIn category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
Further reading
(Downloadable PDF) This give some of the history of groupoids, namely the origins in work of Heinrich BrandtHeinrich Brandt
Heinrich Brandt was a German mathematician who was the first to develop the concept of groupoid....
on quadratic forms, and an indication of later work up to 1987, with 160 references.. A web article with lots of references explaining how the groupoid concept has to 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 at PlanetPhysics.org and Bangor.ac.uk Shows how generalisations of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
lead to Galois groupoids.