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{{About|2-dimensional higher groups|2-primary groups|p-primary group}}
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In [[mathematics]], a '''2-group''', or '''2-dimensional higher group''', is a certain combination of [[Group (mathematics)|group]] and [[groupoid]]. The 2-groups are part of a larger hierarchy of [[n-group (category theory)|''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 module]]s, 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 equivalance]] 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<sub>''B''</sub>(''x''). The objects are the [[automorphism]]s 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<sub>''B''</sub>(''x'') is the only data left in ''B''.  Thus, 2-groups may be identified with one-object 2-groupoids, much as groups may be idenitified 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''<sub>0</sub>. This will not work for arbitrary 2-groups; however, if one identifies isomorphic objects, then the [[equivalence class]]es form a group, called the ''fundamental group'' of ''G'' and written π<sub>1</sub>(''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''<sub>''G''</sub>. The [[automorphism group]] of ''I''<sub>''G''</sub> is an [[abelian group]] by the [[Eckmann–Hilton argument]], written Aut(''I''<sub>''G''</sub>) or π<sub>2</sub>(''G'').
 
The fundamental group of ''G'' [[group action|act]]s on either side of π<sub>2</sub>(''G''), and the associator of ''G'' (as a monoidal category) defines an element of the [[group cohomology|cohomology group]] H<sup>3</sup>(π<sub>1</sub>(''G''),π<sub>2</sub>(''G'')). In fact, 2-groups are [[classification theorem|classified]] in this way: given a group π<sub>1</sub>, an abelian group π<sub>2</sub>, a group action of π<sub>1</sub> on π<sub>2</sub>, and an element of H<sup>3</sup>(π<sub>1</sub>,π<sub>2</sub>), there is a unique ([[up to]] equivalence) 2-group ''G'' with π<sub>1</sub>(''G'') isomorphic to π<sub>1</sub>, π<sub>2</sub>(''G'') isomorphic to π<sub>2</sub>, and the other data corresponding.
 
== 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 Π<sub>2</sub>(''X'',''x''). As a monoidal category, the objects are [[loop (topology)|loop]]s at ''x'', with multiplication given by concatenation, and the morphisms are basepoint-preserving [[homotopy|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 space|pointed]] [[connected space]] whose fundamental 2-group is ''G'' and whose [[homotopy group]]s π<sub>''n''</sub> are trivial for ''n''&nbsp;&gt; 2. In this way, 2-groups [[classification theorem|classify]] pointed connected weak homotopy 2-types. This is a generalisation of the construction of [[Eilenberg–Mac Lane space]]s.
 
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> \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> \pi_{2}(X,x) = \pi_{2}(\Pi_{2}(X,x)) .\!</math>
Thus, both the first and second [[homotopy group]]s of a space are contained within its fundamental 2-group. As this 2-group also defines an action of π<sub>1</sub>(''X'',''x'') on π<sub>2</sub>(''X'',''x'') and an element of the cohomology group H<sup>3</sub>(π<sub>1</sub>(''X'',''x''),π<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.
 
== References ==
* [[John C. Baez]] and Aaron D. Lauda, [http://arxiv.org/abs/math.QA/0307200 Higher-Dimensional Algebra V: 2-Groups], Theory and Applications of Categories 12 (2004), 423–491.
* [[John C. Baez]] and Danny Stevenson, [http://arxiv.org/abs/0801.3843 The Classifying Space of a Topological 2-Group].
* [[Ronald Brown (mathematician)|R. Brown]] and P.J. Higgins, ``The classifying space of a crossed complex'', Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120.
* [[Ronald Brown (mathematician)|R. Brown]], P.J. Higgins, R. Sivera, '' Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids,'' EMS Tracts in Mathematics Vol. 15, 703 pages. (2011).
* Hendryk Pfeiffer, [http://arxiv.org/abs/math/0411468 2-Groups, trialgebras and their Hopf categories of representations], Adv. Math. 212 No. 1 (2007) 62–108.
* [http://ncatlab.org/nlab/show/2-group 2-group] at the [[nLab|''n''-Category Lab]].
 
== External links ==
* 2008 [http://mat.uab.cat/~kock/crm/hocat/cat-groups/ Workshop on Categorical Groups] at the [[Centre de Recerca Matemàtica]]
 
[[Category:Group theory]]
[[Category:Higher category theory]]
[[Category:Homotopy theory]]

Latest revision as of 13:23, 11 January 2015

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