In mathematics, an ** n-group**, or

- Examples
- Eilenberg-Maclane spaces
- 2-groups
- 3-groups
- n-groups
- See also
- References
- Algebraic models for homotopy n-types
- Cohomology of higher groups
- Cohomology of higher groups over a site

The general definition of -group is a matter of ongoing research. However, it is expected that every topological space will have a *homotopy -group* at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group , or the entire Postnikov tower for .

One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group can be turned into an Eilenberg-Maclane space through a simplicial construction,^{ [1] } and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write as , and for an abelian group , is written as .

The definition and many properties of 2-groups are already known. 2-groups can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple where are groups with abelian,

a group morphism, and a cohomology class. These groups can be encoded as homotopy -types with and , with the action coming from the action of on higher homotopy groups, and coming from the Postnikov tower since there is a fibration

coming from a map . Note that this idea can be used to construct other higher groups with group data having trivial middle groups , where the fibration sequence is now

coming from a map whose homotopy class is an element of .

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 3-types of groups.^{ [2] } Essential, these are given by a triple of groups with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this 3-group as a homotopy 3-type , the existence of universal covers gives us a homotopy type which fits into a fibration sequence

giving a homotopy type with trivial on which acts on. These can be understood explicitly using the previous model of -groups, shifted up by degree (called delooping). Explicitly, fits into a postnikov tower with associated Serre fibration

giving where the -bundle comes from a map , giving a cohomology class in . Then, can be reconstructed using a homotopy quotient .

The previous construction gives the general idea of how to consider higher groups in general. For an n group with groups with the latter bunch being abelian, we can consider the associated homotopy type and first consider the universal cover . Then, this is a space with trivial , making it easier to construct the rest of the homotopy type using the postnikov tower. Then, the homotopy quotient gives a reconstruction of , showing the data of an -group is a higher group, or Simple space, with trivial such that a group acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoids ^{ [3] }^{pg 295} since the groupoid structure models the homotopy quotient .

Going through the construction of a 4-group is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume is trivial, so the non-trivial groups are . This gives a postnikov tower

where the first non-trivial map is a fibration with fiber . Again, this is classified by a cohomology class in . Now, to construct from , there is an associated fibration

given by a homotopy class . In principal^{ [4] } this cohomology group should be computable using the previous fibration with the Serre spectral sequence with the correct coefficients, namely . Doing this recursively, say for a -group, would require several spectral sequence computations, at worse many spectral sequence computations for an -group.

For a complex manifold with universal cover , and a sheaf of abelian groups on , for every there exists^{ [5] } canonical homomorphisms

giving a technique for relating n-groups constructed from a complex manifold and sheaf cohomology on . This is particularly applicable for complex tori.

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 the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

**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.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

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. Intuitively, homotopy groups record information about the basic shape, or *holes*, of a topological space.

In mathematics, specifically in homotopy theory, a **classifying space***BG* of a topological group *G* is the quotient of a weakly contractible space *EG* by a proper free action of *G*. It has the property that any *G* principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle *EG* → *BG*. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.

In mathematics, specifically algebraic topology, an **Eilenberg–MacLane space** is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

In the mathematical field of algebraic topology, the **homotopy groups of spheres** describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

In mathematics, the **cohomology operation** concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if *F* is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from *F* to itself. Throughout there have been two basic points:

- the operations can be studied by combinatorial means; and
- the effect of the operations is to yield an interesting bicommutant theory.

In homotopy theory, a branch of algebraic topology, a **Postnikov system** is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

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, 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, a **weak equivalence** is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.

In mathematics, more specifically in homotopy theory, a **simplicial presheaf** is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a **simplicial sheaf** on a site is a simplicial object in the category of sheaves on the site.

In category theory, a branch of mathematics, an **∞-groupoid** is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, **homotopy theory** is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.

In Mathematics, an **Abelian 2-group** is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups.

- ↑ "On Eilenberg-Maclane Spaces" (PDF). Archived (PDF) from the original on 28 Oct 2020.
- ↑ Conduché, Daniel (1984-12-01). "Modules croisés généralisés de longueur 2".
*Journal of Pure and Applied Algebra*.**34**(2): 155–178. doi: 10.1016/0022-4049(84)90034-3 . ISSN 0022-4049. - ↑ Goerss, Paul Gregory. (2009).
*Simplicial homotopy theory*. Jardine, J. F., 1951-. Basel: Birkhäuser Verlag. ISBN 978-3-0346-0189-4. OCLC 534951159. - ↑ "Integral cohomology of finite Postnikov towers" (PDF). Archived (PDF) from the original on 25 Aug 2020.
- ↑ Birkenhake, Christina (2004).
*Complex Abelian Varieties*. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 573–574. ISBN 978-3-662-06307-1. OCLC 851380558.

- Hoàng Xuân Sính, Gr-catégories, PhD thesis, (1973)
- John C. Baez and Aaron D. Lauda, Higher-Dimensional Algebra V: 2-Groups, Theory and Applications of Categories 12 (2004), 423–491.
- David Michael Roberts and Urs Schreiber, The inner automorphism 3-group of a strict 2-group, Journal of Homotopy and Related Structures, vol. 3(1) (2008), pp. 193–245.
- Classification of weak 3-groups
- Stacks and the homotopy theory of simplicial sheaves

- Algebraic invariants for homotopy types
- On Algebraic Models for Homotopy 3-Types
- Computing Homotopy Types Using Crossed N-Cubes of Groups
- Weak units and homotopy 3-types
- Algebraic models for homotopy n-types - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space with values in a higher group , giving higher cohomology groups . If we are considering as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.