# 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 $B\left(x,y\right)$, 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 ${1}_{x}:B\left(x,x\right)$ called the identity morphism or identity $1$-cell at $x$;
• For each triple of $0$-cells $x,y,z$, a functor $\circ :B\left(y,z\right)×B\left(x,y\right)\to B\left(x,z\right)$ called horizontal composition;
• For each pair of $0$-cells $x,y$, a functor ${-}^{-1}:B\left(y,x\right)\to B\left(x,y\right)$ called the inverse operation;
• For each pair of $0$-cells $x,y$, two natural isomorphisms called unitors: ${\mathrm{id}}_{B\left(x,y\right)}\circ {\mathrm{const}}_{{1}_{x}}\cong {\mathrm{id}}_{B\left(x,y\right)}\cong {\mathrm{const}}_{{1}_{y}}\circ {\mathrm{id}}_{B\left(x,y\right)}:B\left(x,y\right)\to B\left(x,y\right)$;
• For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from ${B}_{y,z}×{B}_{x,y}×{B}_{w,x}$ 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 ${\mathrm{id}}_{B\left(x,y\right)}$ 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)