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.

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.

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 Π 2 (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. \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 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 2-type.

Notes

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