2-group

In mathematics, particularly category theory, a is a groupoid with a way to multiply objects and morphisms, making it resemble a group. They are part of a larger hierarchy of.
They were introduced by Hoàng Xuân Sính in the late 1960s under the name, 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.

Strict 2-groups

Much of the literature focuses on strict 2-groups. A strict is a strict monoidal category in which every morphism is invertible and every object has a strict inverse.
A strict 2-group is a group object in a category of categories; as such, they could be called groupal categories. Conversely, a strict 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, in general can be seen as a weakening of crossed modules.
Every 2-group is equivalent to a strict, although this can't be done coherently: it doesn't extend to homomorphisms.

Examples

Given a category C, we can consider the Aut C. This is the monoidal category whose objects are the autoequivalences of 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 of X at x, written Π₂. 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 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 of x in B, written Aut_Bx. 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 and x is its only object, then Aut_Bx is the only data left in B. Thus, may be identified with , much as groups may be identified with one-object groupoids and monoidal categories may be identified with 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 ; however, if one identifies isomorphic objects, then the equivalence classes form a group, called the fundamental group of G and written π₁G.
As a monoidal category, any G has a unit object I_G. The automorphism group of I_G is an abelian group by the Eckmann–Hilton argument, written Aut 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. In fact, 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, there is a unique G with π₁G isomorphic to π₁, π₂G isomorphic to π₂, and the other data corresponding.
The element of H³ associated to a 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 of a topological space X and a point x is the Π₂, 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 G, one can find a unique pointed connected space whose fundamental is G and whose homotopy groups π_n are trivial for n > 2. In this way, 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 of X at x; that is,
This fact is the origin of the term "fundamental" in both of its instances.
Similarly,
Thus, both the first and second homotopy groups of a space are contained within its fundamental. As this also defines an action of π₁ on π₂ and an element of the cohomology group H³, π₂), this is precisely the data needed to form the Postnikov tower of X if X is a pointed connected homotopy 2-type.