2-group

2-dimensional higher groups p-groups with p 2 p-group No. 2 Group RAF In mathematics, particularly category theory, a 2-group is a groupoid with a way to multiply objects and morphisms, 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,Hoàng Xuân Sính Hoàng Xuân Sính http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html Gr-catégories Thesis 1975 https://web.archive.org/web/20150721143805/http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html 2015-07-21 Baez John C. 2023 Hoàng Xuân Sính's thesis: categorifying group theory math.CT 2308.05119 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 (small) 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.March 2019 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 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 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 AutBx. 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 AutBx 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 π1G. (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 IG. The automorphism group of IG is an abelian group by the Eckmann–Hilton argument, written Aut(IG) or π2G. The fundamental group of G acts on either side of π2G, and the associator of G defines an element of the cohomology group H3(π1G, π2G). 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 π1G isomorphic to π1, π2G 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 As mentioned above, the fundamental 2-group of a topological space X and a point x is the 2-group 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 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 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. See also n-group Abelian 2-group Notes References Baez John C. John C. Baez Lauda Aaron D. math.QA/0307200 Higher-dimensional algebra V: 2-groups Theory and Applications of Categories 12 2004 423–491 http://www.tac.mta.ca/tac/volumes/12/14/12-14.pdf Baez John C. John C. Baez Stevenson Danny 0801.3843 The classifying space of a topological 2-group Algebraic Topology. The Abel Symposium 2007 Baas Nils Friedlander Eric Eric Friedlander Jahren Bjørn Østvær Paul Arne Springer, Berlin 1–31 2009 Brown Ronald Ronald Brown (mathematician) Higgins Philip J. The classifying space of a crossed complex Mathematical Proceedings of the Cambridge Philosophical Society 110 1 July 1991 95–120 10.1017/S0305004100070158 1991MPCPS.110...95B Brown Ronald Ronald Brown (mathematician) Higgins Philip J. Sivera Rafael Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids EMS Tracts in Mathematics 15 August 2011 10.4171/083 2841564 1237.55001 math/0407275 978-3-03719-083-8 Pfeiffer Hendryk math/0411468 2-Groups, trialgebras and their Hopf categories of representations Advances in Mathematics 212 1 2007 62–108 10.1016/j.aim.2006.09.014 free Cegarra Antonio Martínez Heredia Benjamín A. Remedios Josué 1003.3820 Double groupoids and homotopy 2-types Applied Categorical Structures 20 323–378 2012 4 10.1007/s10485-010-9240-1 free External links 2-group 2-group 2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica Category:Group theory Category:Higher category theory Category:Homotopy theory