# nLab double groupoid

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

A double groupoid is, equivalently,

Equipped with the relevant extra stuff, structure, property one obtains notions of double topological groupoids, double Lie groupoids, etc.

## Definition

###### Definition

Let $H$ be a (2,1)-topos, hence a (2,1)-category whose objects may be thought of as groupoids equipped with some geometric structure (stacks). Then a double groupoid with that geometric structure is a groupoid object in an (2,1)-category in $H$, hence a simplicial object

${𝒢}_{•}\in {H}^{{\Delta }^{\mathrm{op}}}$\mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}

which satisfies the groupoidal Segal conditions.

In the literature, the following special cases of def. 1 are often taken to be the default notion of “double groupoid”.

###### Example

The archetypical special case of def. 1 is that where $H=$ Grpd is the (2,1)-category of bare (geometrically discrete) groupoids.

###### Example

A special case of example 1, in turn, are bare double groupoids in the image of the embedding ${\mathrm{Grpd}}_{1}^{{\Delta }^{\mathrm{op}}}\to {\mathrm{Grpd}}^{{\Delta }^{\mathrm{op}}}$, where ${\mathrm{Grpd}}_{1}$ is the 1-category of groupoids (suppressing the 2-morphisms given by natural isomorphisms). A groupoid object in ${\mathrm{Grpd}}_{1}$ is equivalently a pair of groupoids ${𝒢}_{1}$ and ${𝒢}_{0}$ equipped with functors $s,t:{𝒢}_{1}\to {𝒢}_{0}$, $i:{𝒢}_{0}\to {𝒢}_{1}$ and $\circ :{𝒢}_{1}{×}_{{𝒢}_{0}}{𝒢}_{1}\to {𝒢}_{1}$ that satisfy the usual axioms of a small category groupoid without any non-trivial natural isomorphisms weakening them. This is called a strict double groupoid.

###### Remark

If one writes out the structure functors

$\begin{array}{c}{𝒢}_{1}\\ {}^{s}↓{↓}^{t}\\ {𝒢}_{0}\end{array}$\array{ \mathcal{G}_1 \\ {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ \mathcal{G}_0 }

of a double groupoid ${𝒢}_{•}$ themselves in components, one obtains a square diagram of the form

$\begin{array}{ccc}{𝒢}_{1,1}& \stackrel{\to }{\to }& {𝒢}_{1,0}\\ ↓↓& & ↓↓\\ {𝒢}_{0,1}& \stackrel{\to }{\to }& {𝒢}_{0,0}\end{array}$\array{ \mathcal{G}_{1,1} & \stackrel{\to}{\to} & \mathcal{G}_{1,0} \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{G}_{0,1} & \stackrel{\to}{\to} & \mathcal{G}_{0,0} }

(where now we are notationally suppressing the degeneracy maps/identity assigning maps, for readability). In this form double groupoid are presented in traditional literature.

###### Example

For $H=$ SmoothGrpd, double groupoids in $H$ which are in the inclusion of LieGrpd${\Delta }^{\mathrm{op}}\to$ SmoothGrpd${}^{{\Delta }^{\mathrm{op}}}$ are called double Lie groupoids.

###### Remark

More generally, one can consider double groupoids in an arbitrary (∞,1)-topos $H$, to be a 3-coskeletal groupoid object in an (∞,1)-category consisting degreewise of 1-truncated objects. The realization map

$\underset{\to }{\mathrm{lim}}:{H}^{{\Delta }^{\mathrm{op}}}\to H$\underset{\to}{\lim} \colon \mathbf{H}^{\Delta^{op}} \to \mathbf{H}

restricted to such double groupoids is a presentation of 2-truncated objects in $H$.

## References

Double Lie groupoids are discussed (usually for the strict case) in

• Ronnie Brown, Kirill Mackenzie, Determination of a double Lie groupoid by its core diagram. J. Pure Appl. Algebra 80 (1992), no. 3, 237–272

• Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge (2005)

Some homotopical aspects of double groupoids and their relationship to homotopy 2-types are explored in

Revised on April 25, 2013 12:33:17 by Urs Schreiber (82.169.65.155)