In mathematics, particularly category theory, a 2-group is a groupoid with a way to multiply objects, making it resemble a group.
They were introduced by Hoàng Xuân Sính in the late 1960s under the name gr-categories,[1][2] and they are also known as categorical groups.
A 2-group is a monoidal category G in which every morphism is invertible and every object has a weak inverse.
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.
As mentioned above, the fundamental 2-group of a topological space X and a point x is the 2-group 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.
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.