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 SÃÂnh in the late 1960s under the name , 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. (Here, a weak inverse of an object x is an object y such that xy and yx are both isomorphic to the unit object.)
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 (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 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.
Given a (small) category C, we can consider the Aut C. This is the monoidal category whose objects are the autoequivalences of C (i.e. equivalences F: CâÂÂC), 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 ÃÂ <sub>2</sub>(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.
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<sub>B</sub>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 (so all objects and morphisms are weakly invertible) and x is its only object, then Aut<sub>B</sub>x 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 G<sub>0</sub>. 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 ÃÂ<sub>1</sub>G. (Note that even for a strict , the fundamental group will only be a quotient group of the underlying group.)
As a monoidal category, any G has a unit object I<sub>G</sub>. The automorphism group of I<sub>G</sub> is an abelian group by the EckmannâÂÂHilton argument, written Aut(I<sub>G</sub>) or ÃÂ<sub>2</sub>G.
The fundamental group of G acts on either side of ÃÂ<sub>2</sub>G, and the associator of G defines an element of the cohomology group H<sup>3</sup>(ÃÂ<sub>1</sub>G, ÃÂ<sub>2</sub>G). In fact, are classified in this way: given a group ÃÂ<sub>1</sub>, an abelian group ÃÂ<sub>2</sub>, a group action of ÃÂ<sub>1</sub> on ÃÂ<sub>2</sub>, and an element of H<sup>3</sup>(ÃÂ<sub>1</sub>, ÃÂ<sub>2</sub>), there is a unique (up to equivalence) G with ÃÂ<sub>1</sub>G isomorphic to ÃÂ<sub>1</sub>, ÃÂ<sub>2</sub>G isomorphic to ÃÂ<sub>2</sub>, and the other data corresponding.
The element of H<sup>3</sup>(ÃÂ<sub>1</sub>, ÃÂ<sub>2</sub>) associated to a is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân SÃÂnh.
As mentioned above, the fundamental of a topological space X and a point x is the ÃÂ <sub>2</sub>(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 G, one can find a unique (up to weak homotopy equivalence) pointed connected space (X,x) whose fundamental is G and whose homotopy groups ÃÂ<sub>n</sub> are trivial for n > 2. In this way, classify pointed connected weak homotopy 2-types. This is a generalisation of the construction of EilenbergâÂÂMac 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 ÃÂ<sub>1</sub>(X,x) on ÃÂ<sub>2</sub>(X,x) and an element of the cohomology group H<sup>3</sup>(ÃÂ<sub>1</sub>(X,x), ÃÂ<sub>2</sub>(X,x)), this is precisely the data needed to form the Postnikov tower of X if X is a pointed connected homotopy 2-type.