# nLab double groupoid

Contents

### 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 $\mathbf{H}$ 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 $\mathbf{H}$, hence a simplicial object

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

which satisfies the groupoidal Segal conditions.

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

###### Example

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

###### Example

A special case of example , in turn, are bare double groupoids in the image of the embedding $Grpd_1^{\Delta^{op}} \to Grpd^{\Delta^{op}}$, where $Grpd_1$ is the 1-category of groupoids (suppressing the 2-morphisms given by natural isomorphisms). A groupoid object in $Grpd_{1}$ is equivalently a pair of groupoids $\mathcal{G}_1$ and $\mathcal{G}_0$ equipped with functors $s,t \colon \mathcal{G}_1 \to \mathcal{G}_0$, $i \colon \mathcal{G}_0 \to \mathcal{G}_1$ and $\circ \colon \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 \to \mathcal{G}_1$ 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

$\array{ \mathcal{G}_1 \\ {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ \mathcal{G}_0 }$

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

$\array{ \mathcal{G}_{1,1} & \stackrel{\to}{\to} & \mathcal{G}_{1,0} \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{G}_{0,1} & \stackrel{\to}{\to} & \mathcal{G}_{0,0} }$

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

###### Remark

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

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

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

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