# nLab bigroupoid

### Context

#### Higher category theory

higher category theory

# 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 $x,y,z,\dots$, also called $0$-cells;
• For each pair of $0$-cells $x,y$, a groupoid $B(x,y)$, whose objects are called morphisms or $1$-cells and whose morphisms are called 2-morphisms or $2$-cells;
• For each $0$-cell $x$, a distinguished $1$-cell $1_x\colon B(x,x)$ called the identity morphism or identity $1$-cell at $x$;
• For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z) \times B(x,y) \to B(x,z)$ called horizontal composition;
• For each pair of $0$-cells $x,y$, a functor ${-}^{-1}\colon B(y,x) \to B(x,y)$ called the inverse operation;
• For each pair of $0$-cells $x,y$, two natural isomorphisms called unitors: $id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y)$;
• For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$; and
• For each triple of $0$-cells $x,y,z$, two natural isomorphisms called the unit and counit between the two composites of ${-}^{-1}$ and $id_{B(x,y)}$ 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)