# 2-group

In mathematics, particularly the branch called category theory, a **2-group** is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of *n*-groups.
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**.

## 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 could be 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^{[citation needed]}.

## Examples

Given a (small) category *C*, we can consider the 2-group 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 2-group** of *X* at *x*, written Π_{2}(*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:^{[3]} one can define a functor on any 2-group *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 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 π_{1}(*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 π_{2}(*G*).

The fundamental group of *G* acts on either side of π_{2}(*G*), and the associator of *G* (as a monoidal category) defines an element of the cohomology group H^{3}(π_{1}(*G*),π_{2}(*G*)). In fact, 2-groups are classified in this way: given a group π_{1}, an abelian group π_{2}, a group action of π_{1} on π_{2}, and an element of H^{3}(π_{1},π_{2}), there is a unique (up to equivalence) 2-group *G* with π_{1}(*G*) isomorphic to π_{1}, π_{2}(*G*) isomorphic to π_{2}, and the other data corresponding.

The element of H^{3}(π_{1},π_{2}) 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

As mentioned above, the fundamental 2-group of a topological space *X* and a point *x* is the 2-group Π_{2}(*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 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,

- [math]\displaystyle{ \pi_{1}(X,x) = \pi_{1}(\Pi_{2}(X,x)) .\! }[/math]

This fact is the origin of the term "fundamental" in both of its 2-group instances.

Similarly,

- [math]\displaystyle{ \pi_{2}(X,x) = \pi_{2}(\Pi_{2}(X,x)) .\! }[/math]

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 π_{1}(*X*,*x*) on π_{2}(*X*,*x*) and an element of the cohomology group H^{3}(π_{1}(*X*,*x*),π_{2}(*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

## Notes

- ↑ Hoàng, Xuân Sính (1975), "Gr-catégories",
*Thesis*, http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html - ↑ Baez, John C. (2023). "Hoàng Xuân Sính's thesis: categorifying group theory". arXiv:2308.05119 [math.CT].
- ↑ Baez Lauda 2004

## References

- "Higher-dimensional algebra V: 2-groups",
*Theory and Applications of Categories***12**: 423–491, 2004, http://www.tac.mta.ca/tac/volumes/12/14/12-14.pdf - Baas, Nils; Friedlander, Eric; Jahren, Bjørn et al., eds. (2009), "The classifying space of a topological 2-group",
*Algebraic Topology. The Abel Symposium 2007*, Springer, Berlin, pp. 1–31 - "The classifying space of a crossed complex",
*Mathematical Proceedings of the Cambridge Philosophical Society***110**(1): 95–120, July 1991, doi:10.1017/S0305004100070158, Bibcode: 1991MPCPS.110...95B -
*Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids*, EMS Tracts in Mathematics,**15**, August 2011, doi:10.4171/083, ISBN 978-3-03719-083-8 - Pfeiffer, Hendryk (2007), "2-Groups, trialgebras and their Hopf categories of representations",
*Advances in Mathematics***212**(1): 62–108, doi:10.1016/j.aim.2006.09.014 - Cegarra, Antonio Martínez; Heredia, Benjamín A.; Remedios, Josué (2012), "Double groupoids and homotopy 2-types",
*Applied Categorical Structures***20**(4): 323–378, doi:10.1007/s10485-010-9240-1

## External links

- 2-group in
*nLab* - 2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica

Original source: https://en.wikipedia.org/wiki/2-group.
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