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. (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 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.

Examples

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'': ''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 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.

Properties

Weak inverses can always be assigned coherently:Baez Lauda 2004 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_''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 (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, 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''₀. 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 π₁''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''_''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'' defines an element of the cohomology group H³(π₁''G'', π₂''G''). 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(π₁, π₂), there is a unique (up to equivalence) ''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 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(''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 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,
: $\pi_1(X,x) = \pi_1(\Pi_2(X,x)) .\!$
This fact is the origin of the term "fundamental" in both of its instances.

Similarly,
: $\pi_2(X,x) = \pi_2(\Pi_2(X,x)) .\!$
Thus, both the first and second homotopy groups of a space are contained within its fundamental . As this 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.

See also

* ''n''-group
* Abelian 2-group

Notes

References

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External links

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* 200
Workshop on Categorical Groups
at the Centre de Recerca Matemàtica

{{Category theory

Group theory
Higher category theory
Homotopy theory