2-group

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.
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 2-group 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 categories; as such, they are also called "groupal categories". Conversely, a strict 2-group 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, 2-groups in general can be seen as a weakening of crossed modules.
Every 2-group is equivalent to a strict 2-group, although this can't be done coherently: it doesn't extend to 2-group homomorphisms.
Properties.
Weak inverses can always be assigned coherently: one can define a functor on any 2-group "G" that assigns a weak inverse to each object and makes that object an adjoint equivalence in the monoidal category "G".
Given a bicategory "B" and an object "x" of "B", there is an "automorphism 2-group" 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 2-groupoid (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, 2-groups may be identified with one-object 2-groupoids, much as groups may be identified with one-object groupoids and monoidal categories may be identified with one-object bicategories.
If "G" is a strict 2-group, then the objects of "G" form a group, called the "underlying group" of "G" and written "G"0. This will not work for arbitrary 2-groups; however, if one identifies isomorphic objects, then the equivalence classes form a group, called the "fundamental group" of "G" and written π1("G"). (Note that even for a strict 2-group, the fundamental group will only be a quotient group of the underlying group.)
As a monoidal category, any 2-group "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 π2("G").
The fundamental group of "G" acts on either side of π2("G"), and the associator of "G" (as a monoidal category) defines an element of the cohomology group H3(π1("G"),π2("G")). In fact, 2-groups 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) 2-group "G" with π1("G") isomorphic to π1, π2("G") isomorphic to π2, and the other data corresponding.
The element of H3(π1,π2) associated to a 2-group is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân Sính.
Fundamental 2-group.
Given a topological space "X" and a point "x" in that space, there is a fundamental 2-group 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.
Conversely, given any 2-group "G", one can find a unique (up to weak homotopy equivalence) pointed connected space "(X,x)" whose fundamental 2-group is "G" and whose homotopy groups π"n" are trivial for "n" > 2. In this way, 2-groups 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 2-group of "X" at "x"; that is,
This fact is the origin of the term "fundamental" in both of its 2-group instances.
Similarly,
Thus, both the first and second homotopy groups of a space are contained within its fundamental 2-group. As this 2-group 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.