2-group

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.

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 modules, and are most often studied in that form. Thus, 2-groups in general can be seen as a weakening of crossed modules.

No. 2 Group RAF

No. 2 Group is a Group of the Royal Air Force which was first activated in 1918, served from 1918–20, from 1936 through the Second World War to 1947, from 1948 to 1958, from 1993 to 1996, was reactivated in 2000, and is today part of Air Command.

The group is currently referred to as the Air Combat Support Group, as it controls the aircraft used to support the Royal Navy and RAF's front line combat force. Assets under command includes the Strategic and Tactical Air Transport aircraft (including VIP/Communication), the RAF Police (including RAF Regiment assets), the Air-to-Air Refuelling aircraft. As from 1 April 2006, with the disbandment of No. 3 Group RAF, it controls also the Airborne Early Warning aircraft, ground based radar installations, Maritime Reconnaissance aircraft and the Search & Rescue Force with Sea King helicopters at six UK coastal bases and Mountain Rescue Teams at four locations, RAF Kinloss, RAF Leuchars, RAF Leeming and RAF Valley. The group is based alongside Air Command at RAF High Wycombe in Buckinghamshire.