In mathematics, particularly category theory, a 2group is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of ngroups. They were introduced by Hoàng Xuân Sính in the late 1960s under the name grcategories,^{[1]}^{[2]} and they are also known as categorical groups.
YouTube Encyclopedic

1/3Views:22 3621 721 51915 526

2. Group Theory

Group theory, abstraction, and the 196,883dimensional monster

Counting in groups  Mathematics  Std.2nd
Transcription
Definition
A 2group 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 2groups
Much of the literature focuses on strict 2groups. A strict 2group 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 2group is a group object in a category of (small) categories; as such, they could be called groupal categories. Conversely, a strict 2group 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, 2groups in general can be seen as a weakening of crossed modules.
Every 2group is equivalent to a strict 2group, although this can't be done coherently: it doesn't extend to 2group homomorphisms.^{[citation needed]}
Examples
Given a (small) category C, we can consider the 2group 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 2group 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 basepointpreserving 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 2group 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 2group 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 2morphisms between these. If B is a 2groupoid (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, 2groups may be identified with oneobject 2groupoids, much as groups may be identified with oneobject groupoids and monoidal categories may be identified with oneobject bicategories.
If G is a strict 2group, then the objects of G form a group, called the underlying group of G and written G_{0}. This will not work for arbitrary 2groups; 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 2group, the fundamental group will only be a quotient group of the underlying group.)
As a monoidal category, any 2group 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 defines an element of the cohomology group H^{3}(π_{1}G, π_{2}G). In fact, 2groups 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) 2group 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 2group is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân Sính.
Fundamental 2group
As mentioned above, the fundamental 2group of a topological space X and a point x is the 2group Π_{2}(X,x), whose objects are loops at x, with multiplication given by concatenation, and the morphisms are basepointpreserving homotopies between loops, with these morphisms identified if they are themselves homotopic.
Conversely, given any 2group G, one can find a unique (up to weak homotopy equivalence) pointed connected space (X,x) whose fundamental 2group is G and whose homotopy groups π_{n} are trivial for n > 2. In this way, 2groups classify pointed connected weak homotopy 2types. 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 2group of X at x; that is,
This fact is the origin of the term "fundamental" in both of its 2group instances.
Similarly,
Thus, both the first and second homotopy groups of a space are contained within its fundamental 2group. As this 2group 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 2type.
See also
Notes
 ^ Hoàng, Xuân Sính (1975), "Grcatégories", Thesis, archived from the original on 20150721
 ^ Baez, John C. (2023). "Hoàng Xuân Sính's thesis: categorifying group theory". arXiv:2308.05119 [math.CT].
 ^ Baez Lauda 2004
References
 Baez, John C.; Lauda, Aaron D. (2004), "Higherdimensional algebra V: 2groups" (PDF), Theory and Applications of Categories, 12: 423–491, arXiv:math.QA/0307200
 Baez, John C.; Stevenson, Danny (2009), "The classifying space of a topological 2group", in Baas, Nils; Friedlander, Eric; Jahren, Bjørn; Østvær, Paul Arne (eds.), Algebraic Topology. The Abel Symposium 2007, Springer, Berlin, pp. 1–31, arXiv:0801.3843
 Brown, Ronald; Higgins, Philip J. (July 1991), "The classifying space of a crossed complex", Mathematical Proceedings of the Cambridge Philosophical Society, 110 (1): 95–120, Bibcode:1991MPCPS.110...95B, doi:10.1017/S0305004100070158
 Brown, Ronald; Higgins, Philip J.; Sivera, Rafael (August 2011), Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics, vol. 15, arXiv:math/0407275, doi:10.4171/083, ISBN 9783037190838, MR 2841564, Zbl 1237.55001
 Pfeiffer, Hendryk (2007), "2Groups, trialgebras and their Hopf categories of representations", Advances in Mathematics, 212 (1): 62–108, arXiv:math/0411468, doi:10.1016/j.aim.2006.09.014
 Cegarra, Antonio Martínez; Heredia, Benjamín A.; Remedios, Josué (2012), "Double groupoids and homotopy 2types", Applied Categorical Structures, 20 (4): 323–378, arXiv:1003.3820, doi:10.1007/s1048501092401
External links
 2group at the nLab
 2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica