mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, nonabelian algebraic topology studies an aspect of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

that involves (inevitably noncommutative)

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. Higher-dimensional categories A f ...

s. Many of the higher-dimensional algebraic structures are

noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

and, therefore, their study is a very significant part of nonabelian

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, and also of Nonabelian Algebraic Topology (NAAT), which generalises to higher dimensions ideas coming from the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

. 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 In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

commonly encountered in classical

. An important part of nonabelian algebraic topology is concerned with the properties and applications of

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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

oids 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 invariant In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

s 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

oids,

crossed module 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, (g,h) \mapsto g \cdot h , and a homomorphism of groups : d\colon H \longrightarro ...

s, crossed complexes and

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

oids are key concepts in developing applications related to homotopy of filtered spaces, higher-dimensional space structures, the construction of the

oid of a

topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...

''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 (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

, as well as categorical and

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 The central object of study in topol ...

. Further examples of such applications include the generalisations of

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...

formalizations of the

noncommutative standard model In theoretical particle physics, the non-commutative Standard Model (best known as Spectral Standard Model ), is a model based on noncommutative geometry that unifies a modified form of general relativity with the Standard Model (extended with ...

s ''via'' fundamental double groupoids and

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

structures even more general than

topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a noti ...

or the lower-dimensional noncommutative spacetimes encountered in several

topological quantum field theories In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...

and noncommutative geometry theories of quantum gravity. A fundamental result in NAAT is the generalised, higher homotopy

van Kampen theorem A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across th ...

proven by R. Brown, which states that ''"the homotopy type of a topological space can be computed by a suitable

colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...

homotopy colimit In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfcon ...

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 In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...

and a topos of topo

Important results in higher-dimensional algebra are also the extensions of the

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

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.MSC(1991): 18D30,11R32,18D35,18D05

References

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Notes

{{reflist Algebraic topology