nLab
double groupoid

Idea

A double groupoid is, equivalently,

• a double category in which all 1-morphisms and 2-morphisms are isomorphisms;

• a groupoid object internal to the 1-category Grpd;

• an n-fold groupoid for $n=2$.

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 $\circ :{𝒢}_1{\times }_{{𝒢}_0}{𝒢}_1\to {𝒢}_1$ 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

$$\begin{array}{c}{𝒢}_1\\ {}^s\downarrow {\downarrow }^t\\ {𝒢}_0\end{array}$$\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

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

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

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

Related concepts

• 2-group

• double Lie algebroid

• 2-category equipped with proarrows

• groupoid object in an (infinity,1)-category

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

• Antonio Martínez Cegarra; Benjamın A. Heredia ; Josué Remedios, Double groupoids and homotopy 2-types Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also arXiv:1003.3820.