In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids.
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Definition[edit]
A 2-group is a monoidal categoryG 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[edit]
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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 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.
Properties[edit]
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 equivalence in the monoidal category G.
Given a bicategoryB and an object x of B, there is an automorphism 2-group of x in B, written AutB(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 AutB(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 G0. 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 π1(G). (Note that even for a strict 2-group, the fundamental group will only be a quotient group of the underlying group.) Tuneup 2 6 3 download free.
As a monoidal category, any 2-group G has a unit object IG. The automorphism group of IG is an abelian group by the Eckmann–Hilton argument, written Aut(IG) or π2(G).
The fundamental group of Gacts on either side of π2(G), and the associator of G (as a monoidal category) defines an element of the cohomology group H3(π1(G),π2(G)). In fact, 2-groups 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) 2-group 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 2-group is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân Sính.
Fundamental 2-group[edit]
Given a topological spaceX and a point x in that space, there is a fundamental 2-group 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.
Conversely, given any 2-group G, one can find a unique (up toweak homotopy equivalence) pointedconnected space(X,x) whose fundamental 2-group is G and whose homotopy groups πn 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,
This fact is the origin of the term 'fundamental' in both of its 2-group instances.
Similarly,
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 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.
References[edit]
John C. Baez and Aaron D. Lauda, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 12 (2004), 423–491.
John C. Baez and Danny Stevenson, The classifying space of a topological 2-group.
R. Brown and P.J. Higgins, The classifying space of a crossed complex, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120.
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, 2-Groups, trialgebras and their Hopf categories of representations, Adv. Math. 212 No. 1 (2007) 62–108.
Hoàng Xuân Sính, Gr-catégories, thesis, 1975.
External links[edit]
Groups Pro 2 12.9
2-group in nLab
2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica
Groups Pro 2 16
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