2-group

Template:About Template:Distinguish In mathematics, particularly category theory, a Template:Nowrap is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of Template:Nowrap. They were introduced by Hoàng Xuân Sính in the late 1960s under the name Template:Nowrap, and they are also known as categorical groups.

Definition

A 2-group is a monoidal category G in which every morphism is invertible and every object has a weak inverse. (Here, a weak inverse of an object x is an object y such that xy and yx are both isomorphic to the unit object.)

Strict 2-groups

Much of the literature focuses on strict 2-groups. A strict Template:Nowrap is a strict monoidal category in which every morphism is invertible and every object has a strict inverse (so that xy and yx are actually equal to the unit object).

A strict 2-group is a group object in a category of (small) categories; as such, they could be called groupal categories. Conversely, a strict Template:Nowrap is a category object in the category of groups; as such, they are also called categorical groups. They can also be identified with crossed modules, and are most often studied in that form. Thus, Template:Nowrap in general can be seen as a weakening of crossed modules.

Every 2-group is equivalent to a strict Template:Nowrap, although this can't be done coherently: it doesn't extend to Template:Nowrap homomorphisms.Template:Cn

Examples

Given a (small) category C, we can consider the Template:Nowrap Aut C. This is the monoidal category whose objects are the autoequivalences of C (i.e. equivalences F: C→C), whose morphisms are natural isomorphisms between such autoequivalences, and the multiplication of autoequivalences is given by their composition.

Given a topological space X and a point x in that space, there is a fundamental Template:Nowrap of X at x, written Π₂(X,x). As a monoidal category, the objects are loops at x, with multiplication given by concatenation, and the morphisms are basepoint-preserving homotopies between loops, with these morphisms identified if they are themselves homotopic.

Properties

Weak inverses can always be assigned coherently: one can define a functor on any Template:Nowrap G that assigns a weak inverse to each object, so that each object is related to its designated weak inverse by an adjoint equivalence in the monoidal category G.

Given a bicategory B and an object x of B, there is an automorphism Template:Nowrap of x in B, written Aut_B x. The objects are the automorphisms of x, with multiplication given by composition, and the morphisms are the invertible 2-morphisms between these. If B is a Template:Nowrap (so all objects and morphisms are weakly invertible) and x is its only object, then Aut_B x is the only data left in B. Thus, Template:Nowrap may be identified with Template:Nowrap Template:Nowrap, much as groups may be identified with one-object groupoids and monoidal categories may be identified with Template:Nowrap bicategories.

If G is a strict 2-group, then the objects of G form a group, called the underlying group of G and written G₀. This will not work for arbitrary Template:Nowrap; however, if one identifies isomorphic objects, then the equivalence classes form a group, called the fundamental group of G and written π₁G. (Note that even for a strict Template:Nowrap, the fundamental group will only be a quotient group of the underlying group.)

As a monoidal category, any Template:Nowrap G has a unit object I_G. The automorphism group of I_G is an abelian group by the Eckmann–Hilton argument, written Aut(I_G) or π₂G.

The fundamental group of G acts on either side of π₂G, and the associator of G defines an element of the cohomology group H³(π₁G, π₂G). In fact, Template:Nowrap are classified in this way: given a group π₁, an abelian group π₂, a group action of π₁ on π₂, and an element of H³(π₁, π₂), there is a unique (up to equivalence) Template:Nowrap G with π₁G isomorphic to π₁, π₂G isomorphic to π₂, and the other data corresponding.

The element of H³(π₁, π₂) associated to a Template:Nowrap is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân Sính.

Fundamental 2-group

As mentioned above, the fundamental Template:Nowrap of a topological space X and a point x is the Template:Nowrap Π₂(X,x), whose objects are loops at x, with multiplication given by concatenation, and the morphisms are basepoint-preserving homotopies between loops, with these morphisms identified if they are themselves homotopic.

Conversely, given any Template:Nowrap G, one can find a unique (up to weak homotopy equivalence) pointed connected space (X,x) whose fundamental Template:Nowrap is G and whose homotopy groups π_n are trivial for n > 2. In this way, Template:Nowrap classify pointed connected weak homotopy 2-types. This is a generalisation of the construction of Eilenberg–Mac Lane spaces.

If X is a topological space with basepoint x, then the fundamental group of X at x is the same as the fundamental group of the fundamental Template:Nowrap of X at x; that is,

${\pi }_1(X,x)={\pi }_1({\mathrm{\Pi }}_2(X,x)).$

This fact is the origin of the term "fundamental" in both of its Template:Nowrap instances.

Similarly,

${\pi }_2(X,x)={\pi }_2({\mathrm{\Pi }}_2(X,x)).$

Thus, both the first and second homotopy groups of a space are contained within its fundamental Template:Nowrap. As this Template:Nowrap also defines an action of π₁(X,x) on π₂(X,x) and an element of the cohomology group H³(π₁(X,x), π₂(X,x)), this is precisely the data needed to form the Postnikov tower of X if X is a pointed connected homotopy 2-type.

See also

• n-group
• Abelian 2-group

References

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External links

• Template:Nlab
• 2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica