nLab
bigroupoid

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Bigroupoids

Idea

A bigroupoid is an algebraic model for (general, weak) 2-groupoids along the lines of a bicategory.

Definition

A bigroupoid is a bicategory in which every morphism is an equivalence and every 2-morphism is an isomorphism.

More explicitly, a bigroupoid consists of:

  • A collection of objects x,y,z,x,y,z,\dots, also called 00-cells;
  • For each pair of 00-cells x,yx,y, a groupoid B(x,y)B(x,y), whose objects are called morphisms or 11-cells and whose morphisms are called 2-morphisms or 22-cells;
  • For each 00-cell xx, a distinguished 11-cell 1 x:B(x,x)1_x\colon B(x,x) called the identity morphism or identity 11-cell at xx;
  • For each triple of 00-cells x,y,zx,y,z, a functor :B(y,z)×B(x,y)B(x,z){\circ}\colon B(y,z) \times B(x,y) \to B(x,z) called horizontal composition;
  • For each pair of 00-cells x,yx,y, a functor 1:B(y,x)B(x,y){-}^{-1}\colon B(y,x) \to B(x,y) called the inverse operation;
  • For each pair of 00-cells x,yx,y, two natural isomorphisms called unitors: id B(x,y)const 1 xid B(x,y)const 1 yid B(x,y):B(x,y)B(x,y)id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y);
  • For each quadruple of 00-cells w,x,y,zw,x,y,z, a natural isomorphism called the associator between the two functors from B y,z×B x,y×B w,xB_{y,z} \times B_{x,y} \times B_{w,x} to B w,zB_{w,z} built out of {\circ}; and
  • For each triple of 00-cells x,y,zx,y,z, two natural isomorphisms called the unit and counit between the two composites of 1{-}^{-1} and id B(x,y)id_{B(x,y)} and the constant functors on the relevant identity morphisms;

such that

Properties

The Duskin nerve operation identifies bigroupoids with 3-coskeletal Kan complexes.

Revised on September 15, 2010 05:25:51 by Urs Schreiber (188.20.66.18)